A quantum channel is the most general operation that can be applied to a
density matrix that creates another physically allowed density matrix. Because
physically allowed density matrices must be positive semi-definite and have
unit trace, this mapping on the space of density matrices must be completely
positive and trace preserving (CPTP). The Kraus representation of a quantum
channel has the form

$\hat{\rho}(t+\tau)=\sum_{\ell=0}^{D_{K}-1}\,\hat{K}^{\phantom{\dagger}}_{\ell}\hat{\rho}(t)\hat{K}^{\dagger}_{\ell},$
(687)

where the different Kraus operators $\hat{K}_{\ell}$ can be unitary or non-
unitary and describe the action of the time evolution of the system under its
own Hamiltonian plus the couplings to the dissipative bath. The number of such
operators $D_{K}$ needed to describe the quantum channel is called the Kraus
dimension (or rank) of the map. For a single qubit $D_{K}\leq 4$, while for a
bosonic mode, the number of operators is formally infinite.

The form of the Kraus map guarantees that it is completely positive. To
guarantee that it is trace-preserving, the Kraus operators must obey the
constraint

$\sum_{\ell=0}^{D_{K}-1}\hat{K}^{\dagger}_{\ell}\hat{K}^{\phantom{\dagger}}_{\ell}=\hat{I},$
(688)

where $\hat{I}$ is the identity operator.

###  B.2 Amplitude Damping

In Section II.4 we briefly introduced the amplitude damping channel for
bosonic modes by using the master equation formalism. Here, we will briefly
discuss its Kraus representation. Following Ref. [318] we can choose to label
the Kraus operators in by the total number of quanta (photons or phonons)
$\ell$ lost during time $\tau$

$\hat{K}_{\ell}=\sqrt{\frac{(1-e^{-\kappa\tau})^{\ell}}{\ell!}}e^{-\frac{\kappa}{2}\tau\hat{n}}a^{\ell};\,\,\ell=0,1,2,3,\ldots$
(689)

where $\hat{n}=a^{\dagger}a$ is the photon number operator and $\kappa$ is the
energy relaxation (photon loss) rate from Eq. (72). We see that the $a^{\ell}$
term in $\hat{K}_{\ell}$ destroys $\ell$ photons, but that even if by chance
no photons are lost, the $\ell=0$ Kraus operator $\hat{K}_{0}$ enacts a ‘no-
jump backaction,’ causing the probability amplitude for Fock states with large
photon numbers to be reduced. This is essentially a Bayesian probability
update – given the additional information that no photons decayed into the
environment, we must revise downward our prior estimate of the initial photon
number.

For small values of $\kappa\tau$, the probability of losing $\ell$ photons in
a short time interval scales as $(\kappa\tau)^{\ell}$ and becomes small for
large $\ell$. Hence we can safely truncate the Kraus dimension to some value
$\ell_{\mathrm{max}}$ if needed. However, the truncated series of Kraus
operators will not be trace-preserving. We can fix this by modifying the ‘no-
jump’ operator $\hat{K}_{0}$ to be

$K_{0}^{\prime}=\sqrt{\hat{I}-\sum_{\ell=0}^{\ell_{\mathrm{max}}}\hat{K}^{\dagger}_{\ell}\hat{K}^{\phantom{\dagger}}_{\ell}}.$
(690)

Because the operator inside the square root is diagonal (and each diagonal
element is non-negative) in the Fock basis, the square root of the matrix can
be found by taking the element-wise square root.

###  B.3 Dephasing

The action of the bosonic dephasing channel can be represented as a sum over
continuously many Kraus operators, each representing a phase-space rotation by
some random angle $\theta$ via

$\mathcal{E}(\rho)=\int_{-\pi}^{\pi}d\theta\ p(\theta)e^{i\theta
a^{\dagger}a}\rho e^{-i\theta a^{\dagger}a}\,.$ (691)

The channel randomizes the phase according to the probability distribution
$p(\theta)$. When $\rho$ is expressed in the photon number basis

$\mathcal{E}(\rho)=\sum_{m,n}\bra{m}\rho\ket{n}\int_{-\pi}^{\pi}d\theta\
p(\theta)e^{i\theta(n-m)}\ket{n}\bra{m}\,,$ (692)

we see that the channel preserves the diagonal elements but reduces the
magnitude of the off-diagonal elements, indicating decoherence.

The exact solution for the quantum capacity of the bosonic dephasing channel
was recently given in [614]. For microwave oscillators realized as three-
dimensional superconducting cavities, intrinsic dephasing rates (i.e., rates
not associated with state changes in the auxiliary qubit coupled to the
cavity) are known to be substantially smaller than the energy relaxation rate
$\kappa$ (unlike superconducting qubits where the rates are often similar) and
have been measured to be below $\kappa_{\phi}\lesssim
1/($10\text{\,}\mathrm{m}\mathrm{s}$)$ [311, 23], and more recently below
$\kappa_{\phi}\lesssim 1/($0.5\text{\,}\mathrm{s}$)$ [25].

In ion traps, the mechanical mode frequencies are sensitive to electric field
noise which can lead to significant dephasing. Similarly, the tweezer beams
trapping neutral atoms may have amplitude or phase fluctuations, which leads
to mode dephasing.

###  B.4 Heating

Microwave resonators are generally in good thermodynamic equilibrium with the
cryogenic refrigeration system, meaning that they are cold enough that the
resonator is rarely thermally excited out of the vacuum state. This is not the
case for ion traps as we will discuss next. Further below, we will also
discuss heating in neutral atom tweezer arrays.

Instead of operating at a typical frequency of GHz as in cQED hardware, the AC
electric field that traps ions in ion traps produces mechanical resonance
frequencies in the MHz regime and they are not generally operated at cryogenic
temperatures. As a result, the dominant quantum noise in the trapped ion
platform is the heating of ions’ motional vibrational modes [615, 616, 15],
which can be described by the time-derivative of the frequency-dependent
average Fock number, i.e., the population of a given mode, $\dot{\bar{n}}$.
The microscopic origin of the heating mechanism remains an open question and
has been historically under debate, such that it is often dubbed ‘anomalous
heating.’

It has been demonstrated that the heating rate depends on the trap frequency
and geometry as well as temperature. Because the ions mostly interact with the
AC trapping electric field, it is clear that the fluctuations in the electric
field can significantly affect the heating rate. From the fluctuation-
dissipation theorem and the fact that the ions couple linearly to the
fluctuating electric field, the heating rate $\dot{\bar{n}}$ can be quantified
by [616]

$\displaystyle\dot{\bar{n}}(\omega,d,T)=\frac{q^{2}}{2m_{I}\hbar\omega}S_{E}(\omega,d,T),$
(693)

where $S_{E}$ and $\omega$ ($\sim$ MHz) are the noise-spectra density and
frequency of the trapping electric field, and $m_{I}$ is the ion’s mass.
Additionally, $S_{E}$ depends on the distance $d$ between the ion and the trap
surface ($\sim$10-100 $\mu m$) as well as the temperature of the operation
$T$, which is found experimentally to obey the following empirical expression
[615]

$\displaystyle S_{E}(\omega,d,T)\propto\omega^{-\alpha}d^{-\beta}f(T),$ (694)

where most experiments suggest $\alpha\sim 1$ and $\beta\sim 4$, $f(T)$ is a
function of temperature that typically exhibits a thermal activation behavior.
The distance dependence $d^{4}$ excludes the possibility of Johnson noise
which should have $d^{2}$ dependence. The reason that $S_{E}$ has such
dependence on $\omega,d,T$ is still under debate, but there have been many
theoretical studies ranging from phenomenological models [617, 618, 619, 620,
621] to _ab initio_ electronic structure and molecular dynamics simulation
[622, 623, 624, 625] to reveal more microscopic details of the heating
mechanism. One possibility is that the molecular absorbates on the trap
surface serve as a thin layer of lossy dielectric medium that induces
dissipation, where the temperature dependence of the motion of the absorbed
molecules can contribute to the temperature scaling. Typical values for the
heating rate $\dot{\bar{n}}$ in Eq. (693) ranges from $\sim 10^{1}$
phonons/sec at 4 K to $10^{3}$ phonons/sec at room temperature, for trapping
frequency $~{}1$ MHz and a trapping distance of $~{}50~{}\mu m$ [626]. Also
note that because the electric field fluctuations have a long wavelength, they
mostly couple to the center of mass mode. Modes other than the center of mass
mode have generally much lower heating rates [615].

In neutral atom arrays, heating is mainly caused by atom movement [8], which
can be reduced by making the atom movement slower. Moreover, in the future,
optimal control movement schemes may be developed to optimize the transport of
oscillator states [627]. Finally, atom movement may be avoided altogether by
using a fixed architecture and using SWAP gates to entangle distant tweezers.

##  Appendix C Bloch-Messiah Decomposition

Recall from the discussion in Sec. III.1, that a unitary corresponding to a
Gaussian transformation on the infinite Hilbert space of $N$ bosonic modes
corresponds to a transformation on the $2N$ mode positions and coordinates (or
equivalently the raising and lowering mode operators) that must be symplectic
to preserve the commutation relations. The Bloch-Messiah decomposition [188,
32, 189] is essentially a singular-value decomposition for these $2N\times 2N$
symplectic matrices. Any $2N\times 2N$ real symplectic matrix $\mathbf{M}$ can
be decomposed into the form

$\displaystyle\mathbf{M}$
$\displaystyle=\mathbf{O_{1}}\mathbf{Z}\mathbf{O_{2}},$ (695)

where $\mathbf{O_{1}}$ and $\mathbf{O_{2}}$ are $2N\times 2N$ orthogonal
symplectic matrices, and $\mathbf{Z}$ is a $2N\times 2N$ diagonal squeezing
matrix containing the singular values in the form

$\displaystyle\mathbf{Z}$
$\displaystyle=\mathrm{Diag}\left(e^{r_{1}},e^{-r_{1}},e^{r_{2}},e^{-r_{2}},\ldots,e^{r_{N}},e^{-r_{N}}\right).$
(696)

Physically this decomposition can be realized by having a system of beam
splitters applied before ($\mathbf{O_{2}}$) and after ($\mathbf{O_{1}}$) a
middle step $\mathbf{Z}$ involving only single-mode squeezing (by varying
amounts on each mode). We will make use of this decomposition in the following
derivations.

There are two reasons that this decomposition is important for experiments.
The first is that it allows synthesis of two-mode squeezing using single-mode
squeezing which is often substantially easier to realize. The second is that
very high-fidelity beam-splitter gates between neighboring qubits now exist
and can be used to synthesize the $\mathbf{O}_{1}$ and $\mathbf{O}_{2}$
operations which in general involve beam splitters between distant resonators.
This in turn requires SWAP networks (again based on beam splitters, see Sec.
V.3.1) to be able to achieve geographically non-local beam splitter gates.

We turn now to some specific examples.

###  C.1 SUM gate

The two-mode SUM gate given in Eq. (309) is a unitary operation that can be
represented by the corresponding $4\times 4$ symplectic matrix (in the mode
operator basis)

$\displaystyle\mathbf{SUM}(\lambda)$
$\displaystyle=\left(\begin{array}[]{cccc}1&0&-\lambda/2&+\lambda/2\\\
0&1&+\lambda/2&-\lambda/2\\\ +\lambda/2&+\lambda/2&1&0\\\
+\lambda/2&+\lambda/2&0&1\end{array}\right),$ (701)

acting on the vector of mode operators

$\displaystyle\left(\begin{array}[]{c}a^{\phantom{{\dagger}}}\\\
a^{\dagger}\\\ b^{\phantom{{\dagger}}}\\\ b^{\dagger}\end{array}\right).$
(706)

The transformation obeys Eq. (172)

$\displaystyle\mathbf{SUM}^{\mathrm{T}}(\lambda)$
$\displaystyle\left(\begin{array}[]{cccc}0&+1&0&0\\\ -1&0&0&0\\\ 0&0&0&+1\\\
0&0&-1&0\end{array}\right)\mathbf{SUM}(\lambda)$ (711) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}0&+1&0&0\\\ -1&0&0&0\\\ 0&0&0&+1\\\
0&0&-1&0\end{array}\right)$ (716)

and is therefore symplectic, proving that it preserves the commutation
relations among the mode operators as required.

The Bloch-Messiah decomposition given in Eq. (313) can be written in this form
using the symplectic matrix representation of the two-mode beam-splitter whose
unitary is given in Eq. (262)

$\displaystyle\mathbf{BS}(\theta,\varphi)=$
$\displaystyle\left(\begin{array}[]{cccc}\cos\frac{\theta}{2}&0&-ie^{i\varphi}\sin\frac{\theta}{2}&0\\\
0&\cos\frac{\theta}{2}&0&ie^{-i\varphi}\sin\frac{\theta}{2}\\\
-ie^{-i\varphi}\sin\frac{\theta}{2}&0&\cos\frac{\theta}{2}&0\\\
0&ie^{+i\varphi}\sin\frac{\theta}{2}&0&\cos\frac{\theta}{2}\end{array}\right).$
(721)

and the symplectic matrix representation of the pair of single-mode squeezers
(with the upper arm of the interferometer being antisqueezed by $s(-r)$ and
the lower arm squeezed by $s(r)$)

$\displaystyle\mathbf{s}(r)\boldsymbol{\otimes}\mathbf{s}(-r)$
$\displaystyle=\left(\begin{array}[]{cccc}\cosh r&+\sinh r&0&0\\\ +\sinh
r&\cosh r&0&0\\\ 0&0&\cosh r&-\sinh r\\\ 0&0&-\sinh r&\cosh
r\end{array}\right).$ (726)

Plugging these into Eq. (313) yields the following expression

$\displaystyle\mathbf{SUM}(\lambda)=\left(\begin{array}[]{cccc}-\cosh(r)\sin(2\theta)&0&-\cosh(r)\cos(2\theta)&\sinh(r)\\\
0&-\cosh(r)\sin(2\theta)&\sinh(r)&-\cosh(r)\cos(2\theta)\\\
\cosh(r)\cos(2\theta)&\sinh(r)&-\cosh(r)\sin(2\theta)&0\\\
\sinh(r)&\cosh(r)\cos(2\theta)&0&-\cosh(r)\sin(2\theta)\end{array}\right).$
(731)

Choosing the parameter values for $\theta$ and $r$ as in Eqs. (314-316) yields
the desired transformation given in Eq. (701).

###  C.2 Two-Mode Squeezing

The two-mode squeezing gate in Box III.5 has the $4\times 4$ symplectic
representation given in Eq. (299)

$\displaystyle\mathbf{TMS}(r,\varphi)=$
$\displaystyle\quad\left(\begin{array}[]{cccc}\cosh(r)&0&0&e^{i\varphi}\sinh(r)\\\
0&\cosh(r)&e^{-i\varphi}\sinh(r)&0\\\ 0&e^{i\varphi}\sinh(r)&\cosh(r)&0\\\
e^{-i\varphi}\sinh(r)&0&0&\cosh(r)\\\ \end{array}\right).$ (736)

It can be synthesized in a similar manner using the same rather than opposite
single-mode squeezing operations in each arm of the interferometer as
illustrated in Fig. 10 and Eq. (303). These squeezing operations are
represented by the symplectic matrix

$\displaystyle\mathbf{s}(r)\mathbf{\otimes}\mathbf{s}(r)$ $\displaystyle=$
$\displaystyle\quad\left(\begin{array}[]{cccc}\cosh r&-\sinh r&0&0\\\ -\sinh
r&\cosh r&0&0\\\ 0&0&\cosh r&-\sinh r\\\ 0&0&-\sinh r&\cosh
r\end{array}\right).$ (741)

Plugging this expression and Eq. (721) into Eq. (303) correctly yields

$\displaystyle\mathbf{TMS}(r,\pi/2)$
$\displaystyle=\left(\begin{array}[]{cccc}\cosh(r)&0&0&i\sinh(r)\\\
0&\cosh(r)&-i\sinh(r)&0\\\ 0&i\sinh(r)&\cosh(r)&0\\\
-i\sinh(r)&0&0&\cosh(r)\\\ \end{array}\right)$ (746)

in agreement with Eq. (736).

The corresponding symplectic transformation $\mathrm{TMS}(r,0)$ of the
quadrature coordinates is

$\displaystyle(x^{\prime}_{a}+x^{\prime}_{b})$
$\displaystyle=e^{+r}(x_{a}+x_{b})$ (747)
$\displaystyle(p^{\prime}_{a}+p^{\prime}_{b})$
$\displaystyle=e^{-r}(p_{a}+p_{b})$ (748)
$\displaystyle(x^{\prime}_{a}-x^{\prime}_{b})$
$\displaystyle=e^{-r}(x_{a}-x_{b})$ (749)
$\displaystyle(p^{\prime}_{a}-p^{\prime}_{b})$
$\displaystyle=e^{+r}(p_{a}-p_{b}),$ (750)

showing that (for large $r$ and the phase choice $\varphi=0$), the position
quadratures of output modes become strongly correlated while the momentum
quadratures become strongly anti-correlated.

##  Appendix D Brief Review of QSP, QSVT Formalism for Qubits

The idea behind quantum signal processing is to use qubit-controlled
operations to perform a polynomial transformation on a $2\times 2$ matrix
[177, 148]. The quantum singular value transformation [429] allows something
more general: it enables a polynomial transformation on the singular values of
a matrix to be performed. Further for QSVT, this matrix need not be a $2\times
2$ matrix; however, the method performs a transformation on the singular
values rather than the eigenvalues which can complicate the procedure by
necessitating block encoding a positive semi-definite version of the
Hamiltonian inside a unitary matrix.

The simplest case of QSP is the case where the matrix is an SU(2) matrix such
as an ordinary single qubit rotation. Specifically, assume that $x\in[-1,1]$
and let $U\in\mathbb{C}^{2\times 2}$ be

$U:=\begin{bmatrix}x&i\sqrt{1-x^{2}}\\\
i\sqrt{1-x^{2}}&x\end{bmatrix}=e^{iX\arccos(x)}$ (751)

The aim of QSP is to find a transformation such that

$\Lambda:U\mapsto\begin{bmatrix}P(x)&i\sqrt{1-x^{2}}Q(x)\\\
i\sqrt{1-x^{2}}Q^{*}(x)&P^{*}(x)\end{bmatrix}$ (752)

for polynomial functions $P(x)$ and $Q(x)$.

Although not all polynomial transformations can be directly constructed using
this approach, the family of attainable transformations is broad.
Specifically, assume that we have a sequence of rotation angles
$\\{\phi_{0},\ldots,\phi_{k}\\}$ such that

$e^{i\phi_{0}Z}\prod_{j=1}^{k}U(x)e^{i\phi_{j}Z}=\begin{bmatrix}P(x)&i\sqrt{1-x^{2}}Q(x)\\\
i\sqrt{1-x^{2}}Q^{*}(x)&P^{*}(x)\end{bmatrix}.$ (753)

Such a set of angles can be found if

  1. 1.

${\rm deg}(P(x))\leq k$ and ${\rm deg}(Q(x))\leq k-1$

  2. 2.

${\rm Parity}(P(x))=k\mod 2$ and ${\rm Parity}(Q(x))=k-1\mod 2$

  3. 3.

$|P(x)|^{2}+(1-x^{2})|Q(x)|^{2}=1$

There exist efficient classical algorithms to find these QSP angles [628, 437,
629, 582, 630]. A generalization of QSP from a single-qubit to a multi-qubit
can be achieved by the qubitization technique. See Ref. [148] for a tutorial
on more details. In the present work, we extend the QSP formulation to bosonic
systems and exploit it for compilation on hybrid oscillator-qubit hardware.

In the main body of the paper, we discuss an alternative convention for QSP
wherein for any $y$

$U\rightarrow U_{z}(y):=\begin{bmatrix}e^{iy}&0\\\ 0&e^{-iy}\end{bmatrix}$
(754)

We can use the previous results to see that the Hadamard transform of the
original matrix can be written as

$\displaystyle U_{\bm{\phi}}$
$\displaystyle=He^{i\phi_{0}Z}\prod_{j=1}^{k}U(x)e^{i\phi_{j}Z}H$
$\displaystyle=e^{i\phi_{0}X}\prod_{j=1}^{k}U_{z}(\cos^{-1}(x))e^{i\phi_{j}X}$
(755)

We then have using the Hadamard transform, for $z=\cos^{-1}(x)$, that

$U_{\bm{\phi}}=\begin{bmatrix}i{\rm Re}(Q(z))\sqrt{1-z^{2}}+{\rm
Re}(P(z))&{\rm Im}(Q(z))\sqrt{1-z^{2}}+i{\rm Im}(P(z))\\\ -{\rm
Im}(Q(z))\sqrt{1-z^{2}}+i{\rm Im}(P(z))&-i{\rm Re}(Q(z))\sqrt{1-z^{2}}+{\rm
Re}(P(z))\end{bmatrix}.$ (756)

We can see from the constraints on $Q$ and $P$ that the norm of each column
and row is

$|P(z)|^{2}+(1-z^{2})|Q(z)|^{2}=1$ (757)

and thus the resulting matrix is manifestly unitary. In this form for the
polynomials $Q(z)$ and $P(z)$, it is clear that in this formalism we can
block-encode a polynomial of degree at most $k$ subject to the above
constraints on $P$ and $Q$.

##  Appendix E Universality of ISAs and Cross Compilation

Here we demonstrate explicit cross-compilation techniques between various
ISAs, including the phase-space and Fock-space ISAs (Sec. E.1), the sideband
and phase-space ISAs (Sec. E.2), as well as the sideband and Fock-space ISAs
(Sec. E.3). These cross-compilation schemes also serve as a universality proof
for all three ISAs on single qubit-oscillator systems, since at least one of
them is universal in previous works [47, 44]. Building upon the single qubit-
oscillator universality, we provide a recursive proof of universality for
multi-qubit-oscillator systems (Sec. E.4) by adding the analytical compilation
of multi-qubit entangling gates as described in Sec. V.3. The cross
compilation schemes and proofs presented in this section are far from optimal,
but we hope the materials presented here will spur more efficient ways for
cross-compilation in future works.

###  E.1 Compilation between the Fock-Space ISA and the Phase-Space ISA

In this section, we explore cross-compilation between the two most important
ISAs in the current paper, the phase-space and the Fock-space ISAs. Sec. E.1.1
provides physical intuition on compiling the SNAP gate from CD gates. Sec.
E.1.2 leverages Hamiltonian simulation techniques to compile the SNAP gate in
the Fock-space ISA using the phase-space ISA. Finally, Sec. E.1.3 presents the
reverse direction of compiling a CD gate using the Fock space ISA.

####  E.1.1 Fock-space ISA from Phase-space ISA: Intuition

SNAP gates (described in Sec. III.4.4) are cavity-controlled qubit rotations
which apply a separate Berry phase to each Fock state in the cavity. These
gates plus unconditional cavity displacements constitute a universal Fock-
space gate set which is dual to the phase-space ISA in the sense that in the
phase-space ISA, it is the cavity operations that are controlled by the qubit
and the qubit rotations are unconditional. As described in Sec. III.4.4),
cavity-controlled qubit operations are most easily engineered in physical
architectures with large dispersive coupling between the ancilla qubit and the
cavity. Conversely the advantage of the phase-space ISA is that its gates can
be rapidly executed (using large displacements) even when the dispersive
coupling is relatively small. An interesting computational task is to attempt
to compile each of these ISAs in terms of the other and analyze the complexity
of this task.

Physically we know how to implement SNAP gates using the strong-dispersive
Hamiltonian (see App. A.1.3)

$H=\chi a^{\dagger}a\sigma_{z}.$ (758)

The phase-space ISA is based on this very Hamiltonian but operates
experimentally with a small value of the dispersive coupling $\chi$. However,
we could use the phase-space ISA with Trotter-Suzuki to synthesize time
evolution under this Hamiltonian with a larger effective value of $\chi$. Then
we can use controlled displacements with this to literally do a quantum
simulation of the SNAP gate dynamics. Although slow, this strategy will in
principle be sufficient to compile a SNAP gate as we outline below.

####  E.1.2 SNAP Gate Compilation by Quantum Hamiltonian Simulation

For this compilation, we will need to use the phase-space ISA to simulate the
dispersive Hamiltonian given in Eq. (758) with the physical dispersive
coupling $\chi$ replaced by a larger effective value, $\chi_{\mathrm{eff}}$.
The time evolution unitary corresponds to a conditional cavity rotation gate

$\mathrm{CR}(\theta)=e^{-i\frac{\theta}{2}\hat{n}\sigma_{z}}$ (759)

which we can synthesize from CD gates via Trotter-Suzuki. Importantly, this
conditional cavity rotation gate can also be interpreted as a rotation of the
qubit around the $z$ axis by an angle that is proportional to the number of
photons in the cavity: $\theta_{n}=\theta\hat{n}$.

The SNAP gate (Sec. III.4.4) is realized within the Fock state ISA via a pair
of SQR gates (Sec. III.4.3). In the phase-space ISA we will make use of a Rabi
drive to rotate the qubit gate

$R_{\varphi}(\delta)=e^{-i\frac{\delta}{2}[\cos\varphi\sigma_{x}+\sin\varphi\sigma_{y}]}.$
(760)

To make the qubit rotation be conditioned on the photon number being $m$, let
us choose $\theta,\delta\ll 1$ and repeatedly execute the gate sequence

$\displaystyle U_{m}(\theta,\delta)$
$\displaystyle=e^{+i\frac{\theta}{2}m\sigma_{z}}\mathrm{CR}(\theta)R_{\varphi}(\delta)$
$\displaystyle=e^{-i\frac{\theta}{2}[\hat{n}-m]\sigma_{z}}R_{\varphi}(\delta).$
(761)

If, for example, we wish to rotate the qubit through angle $\pi$ iff the
photon number is $m$, then we repeat this $\pi/\delta$ times. If the photon
number is not equal to $m$, then the rapid rotation of the qubit about the $z$
axis simulates the strong dispersive coupling that makes the Rabi drive off-
resonant and hence ineffective at rotating the qubit. Thus we have produced
the SQR gate that selectively rotates the qubit conditioned on the photon
number. A second SQR gate with a different Rabi phase $\varphi$ will then
impart the desired phase on cavity Fock state $m$ yielding the SNAP gate.

If we simply rotate the qubit through $2\pi$ conditioned on the photon number
being $m$, we have a SNAP gate that produces a Grover oracle that adds a minus
sign to Fock state $|m\rangle$

$\displaystyle G=\hat{I}-2|m\rangle\langle m|.$ (762)

If we choose special values of $\theta,\delta$ that, rather than being very
small, are rational multiples of $\pi$, we can achieve other novel gates that,
for example, execute a ‘superparity’ operation. Executing the unitary $N$
times with $\delta=\frac{2\pi}{N}$ for selected values of $\theta$ applies a
minus sign to every Fock state satisfying the condition $\hat{n}=0\mod\ell$
shown in Table E.1

$\displaystyle G_{\ell}=\hat{I}-2\sum_{m}|m\ell\rangle\langle m\ell|.$ (763)

All the gates shown in Table E.1 disentangle the ancilla nearly perfectly from
the cavity and leave the ancilla in its starting state. However, the phases
for those Fock states that are supposed to be 0 can be erroneous. The worst-
case phase error is shown the last column. For this simple algorithm, the
error scales towards zero only as $1/N$. There is doubtless a more efficient
Hamiltonian simulation scheme that can do better (though perhaps ruining the
precise modular in $n$ feature of this simple version). Note that the errors
in the QSP realization of the dispersive Hamiltonian evolution have not been
included here.

Table E.1: QSP for ‘superparity’ operations. Applying the unitary $U_{0}(\theta,\delta)$ found in Eq. (761) $N$ times with $m=0$ and $\delta=\frac{2\pi}{N}$, applies a minus sign to every Fock state satisfying the condition shown in the first column. The phases applied to the other states are $0$ to varying degrees of approximation. The worst-case error is shown in the last column and scales to zero like $1/N$. Note that the errors in the QSP realization of the dispersive Hamiltonian evolution have not been included here.  Condition $n=0\mod\ell$ | $\theta$ | $N$ |  Max phase error (radians)
---|---|---|---
$\mod 2$ | $\pi/2$ | $4$ | $0$
$\mod 4$ | $\pi/4$ | $32$ | $0.152$
$\mod 4$ | $\pi/4$ | $64$ | $0.077$
$\mod 4$ | $\pi/4$ | $128$ | $0.0385$
$\mod 8$ | $\pi/8$ | $64$ | $0.184$
$\mod 8$ | $\pi/8$ | $128$ | $0.093$
$\mod 8$ | $\pi/8$ | $256$ | $0.047$
$\mod 16$ | $\pi/16$ | $128$ | $0.192$
$\mod 16$ | $\pi/16$ | $256$ | $0.097$

####  E.1.3 Phase-Space ISA from Fock-Space ISA

We now turn to the problem of compiling the phase-space ISA in terms of the
Fock-space ISA. In particular, this requires the compilation of single-qubit
rotations $R_{\varphi}(\theta)$ and conditional displacements
$\textrm{CD}(\alpha)$ in terms of selective qubit rotations
$\textrm{SQR}(\vec{\theta},\vec{\varphi})$ and unconditional displacements
$D(\alpha)$. Realizing the first of these is particularly straightforward: one
just needs to choose $\theta_{n}=\theta$ and $\varphi_{n}=\varphi$, such that
the same qubit rotation is applied for every Fock state, reducing the SQR gate
to a single qubit rotation $R_{\varphi}(\theta)$.

To synthesize a controlled displacement, we next recognize that an SQR gate
with $\theta_{n}=n\pi$ conjugated by single qubit rotations $R_{y}(\pi/2)$
realizes a controlled parity operation CP (Box III.8). We note that one can
more efficiently realize this gate using the native dispersive interaction,
though here we fix ourselves to gates within the phase-space ISA to prove
cross-compilation capability. With the controlled parity gate as a primitive,
we can now use the hybrid oscillator-qubit gate synthesis technique outlined
in Section V.3.1 [see the text surrounding Eq. (394)] to realize a conditional
displacement,

$\textrm{CD}(\alpha)=\textrm{CP}\ D(i\alpha)\ \textrm{CP}^{\dagger},$ (764)

thus completing the compilation of the phase-space ISA in terms of the Fock-
space ISA.

###  E.2 Compilation between Sideband ISA and Phase-space ISA

In this section, we show that cross compilation between the sideband gates and
the controlled displacement gates is possible via Trotterization.

####  E.2.1 Sideband gates from controlled displacement

We first show that sideband gates can be compiled from controlled
displacements. Redefining $\alpha=-i\frac{\theta}{2}e^{i\phi}$, the red
sideband gate in (682) can be rewritten as

$\displaystyle\text{JC}(\theta,\varphi)$
$\displaystyle=e^{\alpha\sigma^{+}a-\alpha^{*}\sigma^{-}a^{\dagger}}$ (765)
$\displaystyle=\left(e^{\delta\sigma^{+}a-\delta^{*}\sigma^{-}a^{\dagger}}\right)^{L}$
(766)

where we have Trotterized into a product of $L$ smaller sideband gates with
$\delta=\alpha/L$. Each small sideband gate admits the following decomposition

$\displaystyle e^{\delta\sigma^{+}a-\delta^{*}\sigma^{-}a^{\dagger}}$
$\displaystyle=\exp\left[\frac{\sigma_{x}}{2}(\delta
a-\delta^{*}a^{\dagger})+\frac{i\sigma_{y}}{2}(\delta
a+\delta^{*}a^{\dagger})\right]$ $\displaystyle=e^{\frac{\sigma_{x}}{2}(\delta
a-\delta^{*}a^{\dagger})}e^{\frac{i\sigma_{y}}{2}(\delta
a+\delta^{*}a^{\dagger})}+O(|\delta|^{2}),$ (767)

where the final line corresponds to two controlled displacement gates.
Therefore, we can compile a single sideband gate with parameter $\alpha$ into
$2L$ controlled-displacement gates with an error of $L\cdot
O(|\delta|^{2})=O(|\alpha|^{2}/L)$. Note that higher-order product formulas
can be used in Eq. (767) to improve the compilation accuracy.

####  E.2.2 Controlled displacement gates from sideband gates

Next, we show the reverse direction of compiling controlled displacement gates
into sideband gates by again relying on Trotterization and product formulas.
Without loss of generality, any controlled displacement gate can always be
written as $e^{i\sigma_{x}(\alpha^{*}a^{\dagger}+\alpha a)}$ upon conjugation
by single-qubit rotations. In addition, we can express this conditional
displacement as a product of $L$ small conditional displacements via
Trotterization,

$e^{i\sigma_{x}(\alpha^{*}a^{\dagger}+\alpha
a)}=\left(e^{i\sigma_{x}(\delta^{*}a^{\dagger}+\delta a)}\right)^{L},$ (768)

where $\delta=\alpha/L$. Using the identity $\sigma_{x}=\sigma^{+}+\sigma^{-}$
in combination with the first-order product formula, each small conditional
displacement by $\delta$ can be rewritten as

$\displaystyle e^{i\sigma_{x}(\delta a^{\dagger}+\delta
a)}=e^{i\delta\sigma^{+}a+i\delta^{*}\sigma^{-}a^{\dagger}}e^{i\delta^{*}\sigma^{+}a^{\dagger}+i\delta\sigma^{-}a}+O(|\delta|^{2}).$
(769)

We can then note that the first term in Eq. (769) is a red sideband gate as
given in Eq. (682), while the second term is a blue sideband gate that (if not
natively available) can be synthesized via a red sideband gate and single-
qubit rotations,

$\displaystyle
e^{i\delta^{*}\sigma^{+}a^{\dagger}+i\delta\sigma^{-}a}=e^{i\frac{\pi}{2}\sigma_{x}}\left[e^{i\delta^{*}\sigma^{-}a^{\dagger}+i\delta\sigma^{+}a}\right]e^{-i\frac{\pi}{2}\sigma_{x}}.$
(770)

Therefore, we can express “small” controlled displacements by $\delta$ as two
sideband gates interleaved by $\pi$ rotations on the qubit,

$\displaystyle e^{i\sigma_{x}(\delta a^{\dagger}+\delta
a)}=V_{s}(i\delta)e^{i\frac{\pi}{2}\sigma_{x}}V_{s}(i\delta)e^{-i\frac{\pi}{2}\sigma_{x}}+O(|\delta|^{2}).$
(771)

Consequently, a “large” controlled displacement by $\alpha$ can be compiled as
a product of $2L$ red and blue sideband gates interleaved with single-qubit
rotations, with an overall error of $O(|\alpha|^{2}/L)$. We note that the red
sideband gates can be alternatively replaced by blue sideband gates, again at
the expense of additional single-qubit rotations. We also note the analogy
here with controlled displacement gates realized in ion traps by simultaneous
red and blue sideband drives. Here we have demonstrated cross-compilation
using first-order product formulas, but generalization to higher-order
formulas is possible for more favorable error scaling.

###  E.3 Compilation between the Sideband ISA and the Fock-space ISA

In this section, we demonstrate the ability to cross-compile between the
sideband ISA and the Fock-space ISA. We note that having already shown how to
cross-compile between (i) the Fock-space and phase-space ISAs and (ii) the
sideband and phase-space ISAs, it is possible to cross-compile between
sideband and Fock-space ISAs by employing the phase-space ISA as an
intermediary. While this is a valid approach that proves cross-compilation is
possible, in this section we discuss strategies for _direct_ cross-compilation
between the sideband and Fock-space ISAs to provide further insight.

####  E.3.1 Sideband gates from Fock-space ISA

To compile sideband operations in terms of the Fock-space ISA, it is necessary
to synthesize both single-qubit rotations and sideband operations. The first
of these is straightforward: by setting all rotation angles
$\theta_{n}=\theta$ and $\phi_{n}=\phi$ for $\forall~{}n$ to be identical, the
SQR gate $\mathrm{SQR}(\vec{\theta},\vec{\phi})$ (Box III.9) naturally reduces
to a single-qubit rotation,

$\displaystyle e^{-i\sum_{n=0}^{n_{\rm
max}}\frac{\theta}{2}\vec{b}\cdot\vec{\sigma}\ket{n}\bra{n}}$
$\displaystyle=e^{-i\frac{\theta}{2}\vec{b}\cdot\vec{\sigma}\sum_{n}\ket{n}\bra{n}}$
$\displaystyle=e^{-i\frac{\theta}{2}\vec{b}\cdot\vec{\sigma}\otimes
I}=R(\theta,\phi)$ (772)

where $\vec{b}$ is determined by the phase $\phi$.

The remaining task is to compile the sideband gate $\text{JC}(\theta,\varphi)$
in Eq. (682) in terms of SQR gate and unconditional displacement. We can
modify the technique developed in Ref. [44] to achieve this. We begin by
defining a particular SQR gate

$\displaystyle U_{n}(\varphi)$
$\displaystyle=e^{-i\frac{\theta}{2}\sigma_{\varphi}\hat{Q}_{n}},$ (773)
$\displaystyle\hat{Q}_{n}$
$\displaystyle=\sum_{n^{\prime}=0}^{n}|n\rangle\langle n|.$ (774)

Now note the following useful commutators

$\displaystyle\hat{J}_{n}$
$\displaystyle=[(a^{\dagger}-a),\hat{Q}_{n}]=\sqrt{n+1}\big{(}|n+1\rangle\langle
n|+|n\rangle\langle n+1|\big{)}$ (775) $\displaystyle\hat{L}_{n}$
$\displaystyle=[i(a^{\dagger}+a),\hat{Q}_{n}]=i\sqrt{n+1}\big{(}|n+1\rangle\langle
n|-|n\rangle\langle n+1|\big{)}.$ (776)

Using unconditional displacements and these commutation relations we obtain
from BCH for small $\alpha\in\mathbb{R}$

$\displaystyle\Omega_{n}=D(\alpha)U_{n}(0)D(-\alpha)U^{\dagger}_{n}(0)$
$\displaystyle\approx e^{-i\frac{\alpha\theta}{2}\sigma_{x}\hat{J}_{n}},$
(777)
$\displaystyle\Xi_{n}=D(i\alpha)U_{n}\left(\frac{\pi}{2}\right)D(-i\alpha)U^{\dagger}_{n}\left(\frac{\pi}{2}\right)$
$\displaystyle\approx e^{-i\frac{\alpha\theta}{2}\sigma_{y}\hat{L}_{n}}.$
(778)

Using Trotter to add these two exponents for all $n$ yields the blue side band
Hamiltonian evolution operator

$\displaystyle\prod_{n=0}^{n_{\mathrm{max}}}\Omega_{n}\Xi_{n}$
$\displaystyle\approx
e^{-i\alpha\theta\sum_{n=0}^{n_{\mathrm{max}}}\sqrt{n+1}\left(\sigma^{+}|n+1\rangle\langle
n|+\sigma^{-}|n\rangle\langle n+1|\right)}.$ (779)

To realize the red sideband we would use

$\displaystyle\prod_{n=0}^{n_{\mathrm{max}}}\Omega_{n}\Xi^{\dagger}_{n}$
$\displaystyle\approx
e^{-i\alpha\theta\sum_{n=0}^{n_{\mathrm{max}}}\sqrt{n+1}\left(\sigma^{-}|n+1\rangle\langle
n|+\sigma^{+}|n\rangle\langle n+1|\right)}.$ (780)

Each gate in Eqs. (779) and (780) will cost $4n_{\rm max}$ unconditional
displacement gates and $4n_{\rm max}$ SQR gates to synthesize. Having
synthesized unconditional qubit rotations and side band unitaries we have
formally compiled the sideband gate set using only SQR gates and uncontrolled
cavity displacements from the Fock space gate set, without going through an
intermediate compilation into the phase-space gate set. While this compilation
is formally correct and convergent (for a finite cutoff, $n_{\mathrm{max}}$ on
the photon number), the accuracy could be improved by using higher-order
product formulas. Finding a highly efficient cross-compilation with shallow
circuit depth remains an open problem.

####  E.3.2 Fock-space ISA from Sideband gates

To compile the SQR gate sets from the sideband gate sets, we may use the
intermediate constructions in Ref. [47]. For the SQR gate, note that Ref. [47]
constructed arbitrary clean SU(2) carrier rotations in the subspace
$\\{\ket{0,n},\ket{1,n}\\}$ for any Fock level $n$ expanded by the upper and
lower states of the qubit. Therefore, multiplying many such gates together
directly leads to a SQR gate.

For the unconditional displacement gate, note that it is also possible to
realize arbitrary SU(2) rotations between any pair of oscillator Fock levels
[47]. Therefore, using many such gates can immediately realize an
unconditional displacement gate on the oscillators.

###  E.4 Universality of multi-qubit-oscillator systems

In this section, we show that the phase-space ISA is universal on
$n$-qubit-$m$-oscillator systems for $\forall n,m\in\mathbb{N}^{+}$ by
explicitly constructing all possible generators of it as given in Eq. (386)
and (387). The idea is to start from the 1-qubit-1-oscillator case and then
repeatedly couple one qubit at a time to first realize an
$n$-qubit-1-oscillator universality. Next, by repeatedly coupling one
oscillator at a time to the $n$-qubit-1-oscillator system, universality can be
shown successively on an $n$-qubit-$j$-oscillator system for $j=2,3,\cdots,m$
until finally the $n$-qubit-$m$-oscillator universality is achieved. We
outline the details in the following. We note a recent work has demonstrated
universal multi-mode control by coupling a single qubit to the modes using a
conditional NOT displacement gate [599] which is in a similar spirit as the CD
gate defined in Box III.7.

####  E.4.1 From $1$-qubit-$1$-oscillator to $n$-qubit-$1$-oscillator
universality

Consider an arbitrary generator on an $n$-qubit-$1$-oscillator system of the
form $G_{Q}h(\hat{x},\hat{p})$ where
$h(\hat{x},\hat{p})=\sum_{r,s}c_{r,s}\hat{x}^{r}\hat{p}^{s}+\mathrm{h.c.}$
acts on the oscillator and $G_{Q}$ acts on the qubits. One way to construct
the qubit generators is via tensor product of individual Pauli matrices
$\bigotimes_{j=1}^{n}\sigma_{j,\alpha_{j}}$ for $\alpha_{j}=0,x,y,z$ where
$\sigma_{j,0}$ is the identity matrix for the $j$th qubit. However, a better
set of generators are the generalized Gell-Mann matrices (GGMs) [631, 47]
which possess cleaner commutation relationships. In the following proof, the
notation $G_{Q}$ will be reserved for the GGMs. Since $h(\hat{x},\hat{p})$ can
be written as a sum of different terms, we only need to focus on generating
one such term, say
$(\bigotimes_{j=1}^{n}\sigma_{j,\alpha_{j}})\hat{x}^{r}\hat{p}^{s}$ for
$\forall j,\alpha_{j},r,s$.

From the 1-qubit-1-oscillator universality, we can first compile the generator
$\sigma_{1,\alpha_{1}}\hat{x}^{r}\hat{p}^{s}$ using qubit rotations on qubit 1
and conditional displacements. The qubit-qubit entangling gate in Sec. V.3 can
be used to first entangle a second qubit with qubit 1 to obtain the generator
$(\sigma_{1,\alpha_{1}}\otimes\sigma_{2,\alpha_{2}})\hat{x}^{r}\hat{p}^{s}$
(proper single-qubit rotation on qubit 2 is needed to precisely realize the
desired Pauli $\sigma_{2,\alpha_{2}}$). This process can be continued to
entangle more qubits one by one until we achieve the desired
$(\bigotimes_{j=1}^{n}\sigma_{j,\alpha})\hat{x}^{r}\hat{p}^{s}$. A linear
combination of these will then give us the desired generator
$G_{Q}\hat{x}^{r}\hat{p}^{s}$.

####  E.4.2 From $n$-qubit-$1$-oscillator to $n$-qubit-$m$-oscillator
universality

Now we consider entangling more oscillators with the above
$n$-qubit-$1$-oscillator system to create the desired generators. Let us first
note the following identity when commuting products of high-powers of
$\hat{x}$ and $\hat{p}$

$\displaystyle\hat{p}^{s}\hat{x}^{r}=\hat{x}^{r}\hat{p}^{s}-is\sum_{u=0}^{r-1}\hat{x}^{r-1-u}\hat{p}^{s-1}\hat{x}^{u}.$
(781)

Recursively applying Eq. (781) to terms on the RHS to push any powers of
$\hat{p}$ to the rightmost position, it can be seen that
$\hat{p}^{s}\hat{x}^{r}$ can be written as a linear combination of
$\hat{x}^{r^{\prime}}\hat{p}^{s^{\prime}}$ where the powers $r^{\prime}$ and
$s^{\prime}$ will be no larger than $r$ and $s$.

Now consider adding one more oscillator to achieve an arbitrary generator on
the $n$-qubit-$2$-oscillator system
$G_{Q}h(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2})$ where
$h(\hat{x}_{1},\hat{p}_{1},\hat{x}_{2},\hat{p}_{2})$ can be written in the
form of Eq. (387). Again, we only need to focus on one term in the series sum
of (387), say
$G_{Q}(\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}\hat{x}_{2}^{r_{2}}\hat{p}_{2}^{s_{2}}+\mathrm{h.c.})$.
Our goal is to generate this from the $n$-qubit-$1$-oscillator generator with
additional gates in the phase-space ISA.

Due to the closure of the $n$-qubit generator set, it is always possible to
find another two elements $G_{Q}^{\prime}$ and $G_{Q}^{\prime\prime}$ such
that $[G_{Q}^{\prime},G_{Q}^{\prime\prime}]=iG_{Q}$. Now consider the
following four Hermitian generators for small $t$

$\displaystyle h_{1}$
$\displaystyle=tG_{Q}^{\prime}(\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}+\mathrm{h.c.}),$
(782) $\displaystyle h_{2}$
$\displaystyle=tG_{Q}^{\prime\prime}(\hat{x}_{2}^{r_{2}}\hat{p}_{2}^{s_{2}}+\mathrm{h.c.}),$
(783) $\displaystyle h_{3}$
$\displaystyle=itG_{Q}^{\prime}(\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}-\mathrm{h.c.}),$
(784) $\displaystyle h_{4}$
$\displaystyle=itG_{Q}^{\prime\prime}(\hat{x}_{2}^{r_{2}}\hat{p}_{2}^{s_{2}}-\mathrm{h.c.}),$
(785)

each of which can be realized following from the $n$-qubit-$1$-oscillator
universality. More specifically, $h_{1}$ and $h_{3}$ can be realized from
universality on $n$-qubits with oscillator 1, whereas $h_{2}$ and $h_{4}$
follow from universality on $n$-qubits with oscillator 2. Using product
formulas we have

$\displaystyle(e^{ih_{1}}e^{ih_{2}}e^{-ih_{1}}e^{-ih_{2}})(e^{ih_{4}}e^{ih_{3}}e^{-ih_{4}}e^{-ih_{3}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}\approx\left(e^{-it^{2}G_{Q}(\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}\hat{x}_{2}^{r_{2}}\hat{p}_{2}^{s_{2}}+\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}\hat{p}_{2}^{s_{2}}\hat{x}_{2}^{r_{2}}+\mathrm{h.c.})}\right)~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(e^{-it^{2}G_{Q}(\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}\hat{x}_{2}^{r_{2}}\hat{p}_{2}^{s_{2}}-\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}\hat{p}_{2}^{s_{2}}\hat{x}_{2}^{r_{2}}+\mathrm{h.c.})}\right)$
$\displaystyle\approx~{}e^{-i2t^{2}G_{Q}(\hat{x}_{1}^{r_{1}}\hat{p}_{1}^{s_{1}}\hat{x}_{2}^{r_{2}}\hat{p}_{2}^{s_{2}}+\mathrm{h.c.})}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(786)

where the generator on the RHS is the desired generator on the
$n$-qubit-2-oscillator system.

More generally, to construct a desired generator on an
$n$-qubit-$m^{\prime}$-oscillator system $(m^{\prime}>2)$ given by the form
$G_{Q}(\prod_{j=1}^{m^{\prime}}\hat{x}_{j}^{r_{j}}\hat{p}_{j}^{s_{j}}+\mathrm{h.c.})$,
we use the universality of an $n$-qubit-$(m^{\prime}-1)$-oscillator to obtain

$\displaystyle h_{1}$
$\displaystyle=tG_{Q}^{\prime}\left(\prod_{j=1}^{m^{\prime}-1}\hat{x}_{j}^{r_{j}}\hat{p}_{j}^{s_{j}}+\mathrm{h.c.}\right)$
(787) $\displaystyle h_{3}$
$\displaystyle=itG_{Q}^{\prime}\left(\prod_{j=1}^{m^{\prime}-1}\hat{x}_{j}^{r_{j}}\hat{p}_{j}^{s_{j}}-\mathrm{h.c.}\right)$
(788)

and use the $n$-qubit-1-oscillator universality for oscillator $m^{\prime}$ to
obtain

$\displaystyle h_{2}$
$\displaystyle=tG_{Q}^{\prime\prime}(\hat{x}_{m^{\prime}}^{r_{m^{\prime}}}\hat{p}_{m^{\prime}}^{s_{m^{\prime}}}+\mathrm{h.c.})$
(789) $\displaystyle h_{4}$
$\displaystyle=tG_{Q}^{\prime\prime}(\hat{x}_{m^{\prime}}^{r_{m^{\prime}}}\hat{p}_{m^{\prime}}^{s_{m^{\prime}}}-\mathrm{h.c.}).$
(790)

Combining Eqs. (787)-(790), and using the same product formula as in Eq.
(786), we obtain

$\displaystyle(e^{ih_{1}}e^{ih_{2}}e^{-ih_{1}}e^{-ih_{2}})(e^{ih_{4}}e^{ih_{3}}e^{-ih_{4}}e^{-ih_{3}})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}\approx
e^{-2it^{2}G_{Q}(\prod_{j=1}^{m^{\prime}}\hat{x}_{m^{\prime}}^{r_{m^{\prime}}}\hat{p}_{m^{\prime}}^{s_{m^{\prime}}}+\mathrm{h.c.})}.$
(791)

This accomplishes the induction step of adding one oscillator. When
$m^{\prime}=m$, we therefore have succeeded in constructing an arbitrary
$n$-qubit-$m$-oscillator generator
$G_{Q}(\prod_{j=1}^{m^{\prime}}\hat{x}_{m^{\prime}}^{r_{m^{\prime}}}\hat{p}_{m^{\prime}}^{s_{m^{\prime}}}+\mathrm{h.c.})$
as given in Eq. (386) for any term in Eq. (387).

##  Appendix F Compilation of Bosonic Operators into and from Standard Qubit
Gate Sets

In Sec. III.6 we mentioned that simulating bosonic hardware is in general
challenging on qubit hardware and presented the CNOT gate count for simulating
a single displacement operator on qubits. In this section, we present the full
details on how such a gate count estimation is obtained. For a comparison of
different mappings, we refer the reader to Ref. [166].

The destruction operator $a$ is a natural operator for bosonic systems. Let us
consider its matrix elements in the qubit representation. For example, the
simple Fock basis expression

$a|8\rangle=\sqrt{8}|7\rangle$ (792)

tells us that applying the destruction operator to the state with 8 bosons
yields the state with 7 bosons. This is equally simple on the standard multi-
qubit basis as well

$a|\psi_{8}\rangle=\sqrt{8}|\psi_{7}\rangle.$ (793)

This should be all one needs to tell a qubit compiler [by giving it the matrix
representation of $a$ in Eq. (63)]. However it is important to note $a$
becomes a non-trivial multi-qubit operation in the binary basis that labels
the states of the individual qubits, for example:

$a|1000\rangle=\sqrt{8}|0111\rangle$ (794)

Thus $a$ is an unnatural weight-4 operator (in this particular example) for
qubits because it flips all four qubits. The full matrix representation of the
destruction operator in terms of direct products of single-qubit operators is
complex because computing which bits have to be flipped requires knowing all
the bits. For example, the particular term in $a$ shown in Eq. (794) is given
in terms of qubit operators by a complicated (and difficult to physically
implement) weight-8 expression

$X_{3}X_{2}X_{1}X_{0}[\hat{I}_{3}-Z_{3}][\hat{I}_{2}+Z_{2}][\hat{I}_{1}+Z_{1}][\hat{I}_{0}+Z_{1}]/16,$
(795)

and there are 15 other similar terms needed to fully represent the operator.
From this, we see immediately the huge potential efficiency of having boson
degrees of freedom available natively in the hardware, if we are interested in
simulating a physical system with bosonic modes.

Conversely, if we would like to use resonators to replace multiple qubits in a
quantum simulator or computer, we will need to reckon with the fact that
simple qubit gates typically correspond to non-trivial unitaries acting on the
oscillator. As we will discuss further below, oscillators coupled to transmon
qubits are ‘controllable’ in the formal language of quantum control theory
[168] and recent experiments have demonstrated an impressive level of control
of such oscillators [168, 48, 632, 77, 49, 51].

A natural alternative to the Fock-Binary encoding is the Gray
Code161616https://en.wikipedia.org/wiki/Gray$\\_$code which is a permutation
of the binary encoding with the property that only one bit flips at a time as
the number the code represents increases or decreases by 1. This would seem to
eliminate the multi-qubit flips we saw in Eq. (794). However which single bit
flips is a complicated function of all the bits so we still have a high-weight
operator and there is no simplification to be gained by using the Gray code.
Hence, we will restrict our attention to the binary code for simplicity.

The implementation of Trotter steps in Hamiltonian simulation on a qubit-based
quantum computer requires mapping of the bosonic Hamiltonian to the qubit
spin-subspace, mapping of the bosonic state to the qubits, and then mapping
the resulting state back to a bosonic state (during classical post-processing)
for interpretation of the results.

Within the context of Trotter-Suzuki-based simulations, one of the most
efficient ways of simulating bosonic systems in qubits is via the
decomposition of the Hamiltonian into a sum of one-sparse Hamiltonians. One-
sparse Hamiltonians are arguably the broadest family of quantum Hamiltonians
that can be directly compiled into quantum circuits with nominally zero error.
These Hamiltonians correspond to matrices that have at most one non-zero
matrix element in every row and column. Most of the common matrices that we
use in quantum computing are one-sparse, including the Pauli operators, single
qubit $Z$-rotations (S, P, T gates), controlled-NOT, controlled-Z, SWAP, and
Toffoli – but not the Hadamard gate.

In our bosonic ISAs, however, many of the gates are not one-sparse when
compiled into qubit-only hardware. In the following, we calculate the number
of qubit gates required to implement a single bosonic displacement operation
in a scalable fashion in an all-qubit device.

Assuming that a boson number cutoff of $K=2^{n}-1$ is used, our strategy is to
encode such a bosonic state using $n$ qubits while using one additional qubit
to ease the calculation of the square root factor from bosonic dynamics. As we
will see in the following, the additional qubit allows us to encode
coefficients of adjacent Fock levels into it for easy computation of the
square root operation as multi-qubit controlled single-qubit rotations.

The intuition of why the additional qubit helps to compute the square root is
as follows. For the displacement operator, the Hamiltonian of interest is

$\displaystyle a+a^{\dagger}$
$\displaystyle=\left[\begin{array}[]{ccccc}0&1&0&0&\\\ 1&0&\sqrt{2}&0&\\\
0&\sqrt{2}&0&\sqrt{3}&\cdots\\\ 0&0&\sqrt{3}&0&\\\
&&\vdots&&\ddots\end{array}\right]$ (801)

which we would like to encode on qubits. Observing the self-repeating (though
overlapping) $2\times 2$ block structure

$\displaystyle\begin{bmatrix}0&\sqrt{k}\\\ \sqrt{k}&0\end{bmatrix}$ (802)

and that, for example

$\displaystyle X\oplus\sqrt{3}X=\left[\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&0&\sqrt{3}\\\ 0&0&\sqrt{3}&0\end{array}\right],$ (807)

we can write $a^{\dagger}+a$ as follows:

$\displaystyle a+a^{\dagger}$
$\displaystyle=[0]\oplus\bigoplus_{j=1}^{\infty}\begin{bmatrix}0&\sqrt{2j}\\\
\sqrt{2j}&0\end{bmatrix}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\bigoplus_{j=0}^{\infty}\begin{bmatrix}0&\sqrt{2j+1}\\\
\sqrt{2j+1}&0\end{bmatrix}$ (808) $\displaystyle=H_{\rm even}+H_{\rm odd}.$
(809)

Here, the “even” and “odd” subscripts denote the parity of the value inside
the square-root. We can therefore simulate the quantum dynamics using gates
analogous to $R_{0}(-2it\sqrt{2j})$ or $R_{0}(-2it\sqrt{2j+1})$.

In the following, we describe in detail the steps necessary to use the above
idea to simulate the displacement operator on qubits, the overview of which is
given in Algorithm 1. We will denote bosonic Fock states by
$|k\rangle_{\mathrm{B}}$ where the non-negative integer $k$ is the boson
number. The corresponding representation of this Fock state in the $n$ qubit
register will be denoted $\ket{{\rm binary}(k)}$ without the subscript
$\mathrm{B}$.

Input : A bosonic state defined in the Fock basis by
$\ket{\psi}_{B}=\sum_{k=0}^{K}a_{k}\ket{k}_{B}$ and an $n$-qubit register
augmented by a single additional qubit (hereafter called the main qubit),
prepare the initial state on the register of $n$ qubits as
$\left(\sum_{k=0}^{K}a_{k}\ket{\mathrm{binary}(k)}\right)\otimes\ket{0}$, a
displacement $\alpha\in\mathbb{R}$ and an error tolerance $\epsilon$

Output : A quantum state $V(\alpha)\ket{\psi_{b}}$ such that
$\|V(\alpha)-D(\alpha)\|\leq\epsilon$

1 Subdivide Hamiltonian into $H=H_{\rm even}+H_{\rm odd}$;

2 Provide a Trotter decomposition of the form
$V(\alpha)=\prod_{j=1}^{N_{\exp}/2}e^{-i\alpha H_{\rm
odd}\tau_{2j-1}}e^{-i\alpha H_{\rm even}\tau_{2j}}$ such that
$\|V(\alpha)-D(\alpha)\|\leq\epsilon/2$;

3 for  _$e^{-iH_{p}t_{p}}\in{\rm terms\ of\ }V(\alpha)$ :_ do

4 Simulate $e^{-iH_{p}t_{p}}$ by transferring the quantum amplitudes to the
main qubit (parameters given for $H_{\rm odd}$):
$\sum_{j=0}^{(K-1)/2}\ket{2j+1}\left(a_{2j}\ket{0}+a_{2j+1}\ket{1}\right)$ for
odd cutoff $K$ ;

5 Compute square-roots using reversible operations within error
$\epsilon/(2N_{\exp}\alpha)$;

6 Act with a series of register-controlled ancilla rotations to implement
Hamiltonian;

7 Transfer amplitudes back to the qubit register.

Algorithm 1 Fock-Binary Simulation of Bosonic Displacement Gate with Qubits

Mapping the bosonic state to two-dimensional subspaces and back. Let us write
the truncated bosonic state as a sum over even and odd Fock states:

$\ket{\psi}_{\mathrm{B}}=\sum_{j=0}^{(K-1)/2}\left(a_{2j}\ket{2j}_{\mathrm{B}}+a_{2j+1}\ket{2j+1}_{\mathrm{B}}\right).$
(810)

First, we assume that it is possible to encode this truncated bosonic state
onto a register of $n$ qubits using a binary representation of the Fock states

$\displaystyle\ket{\psi}=\sum_{j=0}^{(K-1)/2}\left(\rule{0.0pt}{10.33327pt}a_{2j}\ket{{\rm
binary}(2j)}+a_{2j+1}\ket{{\rm binary}(2j+1)}\right).$ (811)

For example, an arbitrary state with a maximum of three bosons

$\displaystyle\ket{\psi}_{\mathrm{B}}$
$\displaystyle=\alpha\ket{0}_{\mathrm{B}}+\beta\ket{1}_{\mathrm{B}}+\gamma\ket{2}_{\mathrm{B}}+\delta\ket{3}_{\mathrm{B}}$
(812)

would require a register of $n=2$ qubits to represent it:

$\displaystyle\ket{\psi}$
$\displaystyle=\alpha\ket{00}+\beta\ket{01}+\gamma\ket{10}+\delta\ket{11}.$
(813)

The preparation of this qubit state is in general not a trivial task; however,
we do not consider the complexity of this task here.

Second, by coupling one additional (‘main’) qubit in the last position, we
store the coefficients on the main qubit:

$\displaystyle\ket{\psi^{\prime}}$
$\displaystyle=\sum_{j=0}^{(K-1)/2}\ket{\mathrm{binary}(2j+1)}\otimes\ket{\psi^{\prime}_{2j+1}}$
(814) $\displaystyle\ket{\psi^{\prime}_{2j+1}}$ $\displaystyle\equiv
a_{2j}\ket{0}+a_{2j+1}\ket{1},$ (815)

such that we can then evolve the system under the displacement Hamiltonian
using a single unitary acting in parallel on all $2\times 2$ subspaces, each
labeled by the projector $\ket{2j+1}\bra{2j+1}$

$\sum_{j=0}^{(K-1)/2}\ket{2j+1}\bra{2j+1}\otimes R_{0}(-i2t\sqrt{2j+1}),$
(816)

where the first part of the tensor product acts on the $n$ qubit register and
the $R_{0}$ gate (Table A.1) rotates the main qubit around the $x$ axis.

To prepare $\ket{\psi^{\prime}}$ in Eq. (815) from $\ket{\psi}\otimes\ket{0}$
for $\ket{\psi}$ given in Eq. (811), we use the knowledge that in binary, an
odd number will end in 1: we use a CNOT gate whose target is the main qubit
and whose control is the least significant bit (LSB) in the $n$-qubit
register. This will flip the main qubit if the state in the first $n$-qubit
(system register) represents an odd state. We obtain

$\displaystyle a\ket{2j}\ket{0}+b\ket{2j+1}\ket{0}$ $\displaystyle\mapsto
a\ket{2j}\ket{0}+b\ket{2j+1}\ket{1}.$ (817)

Applying an incrementer operation on the system register, controlled on the
main qubit being in $\ket{0}$, we obtain

$a\ket{2j}\ket{0}+b\ket{2j+1}\ket{1}\mapsto\ket{2j+1}(a\ket{0}+b\ket{1}).$
(818)

For the example initial state in Eq. (813), we now have:

$\ket{01}\left(\alpha\ket{0}+\beta\ket{1}\right)+\ket{11}\left(\gamma\ket{0}+\delta\ket{1}\right)$
(819)

In the following, we calculate the two-qubit entangling gates needed to
operate Eqs. (817) and (818) for encoding the coefficients to the main qubit.
We need one CNOT gate and a single-qubit controlled $n$-qubit incrementer.

An incrementer, in which we do not reset the additional carry qubits required
for the computation of the incrementation, takes $(n-2)$ Toffoli gates and $n$
CNOT gates (with an $n-1$ qubit overhead) [633]. There are multiple ways to
implement Toffoli gates, but the standard decomposition uses $6$ CNOT gates
[1]. Assuming that decomposition, an incrementer takes $7n-12$ CNOT gates.

To implement a controlled incrementer, we can use an extra register of qubits:
we carry out a controlled SWAP between the system register and the extra
register, then implement the controlled adder on the extra register, and carry
out another controlled SWAP to return the incremented value to the system
register. A controlled SWAP gate between the two registers requires $n$
Toffoli and $2n$ CNOT gates. This is because a SWAP gate between qubits $i$
and $j$ can be implemented using ${\rm CNOT}_{ij}{\rm CNOT}_{ji}{\rm
CNOT}_{ij}$ and this circuit can be transformed into a controlled SWAP by only
controlling the ${\rm CNOT}_{ji}$ gate. This implies that two CNOTs and one
Toffoli are needed to do a controlled SWAP, which corresponds to $8$ CNOT
gates. Therefore, with the given construction, a controlled incrementer
requires $23n-12$ CNOT gates.

Because we need to map the amplitudes to the main qubit to make the
computation and back to the system register to interpret the results, this
requires two controlled-NOT gates, one controlled incrementer and one
controlled decrementer, and the aforementioned controlled rotation for the
implementation of the Hamiltonian. The complexity of a decrementer is the same
as that of an adder (up to single qubit gates).

Therefore, for this construction, the total number of CNOT gates required to
transfer two neighboring Fock state amplitudes from the bosonic state
(represented in binary in an $n$ qubit register) onto the main qubit, and
return that information to the binary qubit register representing the Fock
state is

$\displaystyle N_{\textrm{CNOT/transfer}}$ $\displaystyle=2\times(23n-12+1)$
$\displaystyle=46n-22.$ (820)

The system register requires $n$ qubits, where $n$ is the number of qubits
required to represent the Fock state up to Fock level $K$ in binary. Although
all amplitudes can be stored in a single qubit, each incrementer requires
$n-1$ extra qubits, and each controlled SWAP operation requires $n$ extra
qubits. We also count the main qubit. Therefore a total of $n+n-1+n+1=3n$
qubits are needed.

Computation of the square-roots. One of the main costs regarding the
entangling gate counts for computation of the dynamics relates to the
implementation of the $R_{0}(-i2t\sqrt{2j+1})$ gate, which maps

$\displaystyle\ket{\mathrm{binary}(2j+1)}\ket{\psi_{2j+1}^{\prime}}$
$\displaystyle\mapsto$
$\displaystyle\ket{\mathrm{binary}(2j+1)}R_{0}(-i2t\sqrt{2j+1})\ket{\psi_{2j+1}^{\prime}}.$
(821)

Similarly, we can address the states within the even subspaces by computing
instead $\ket{{\rm binary}(2j)}\\}$ Since the square root needs to be computed
for each different value of $2j$ and $2j+1$ in the superposition, we cannot
construct a fixed circuit for it. It would be possible to calculate the square
roots classically in advance and store them in a register, however, given the
number of values we will need in cases with large cutoffs, the cost may become
prohibitive. Several approaches are commonly taken to implement functions such
as this. The first approach directly converts classical networks for these
operations into quantum gates [634] through the use of the Bennett trick. This
approach is time-consuming, but more importantly, often requires hundreds of
qubits to maintain the result.

The cost of performing the square root transformation can be determined from
the cost of a reciprocal square root operation. The reciprocal square root can
be readily computed using Newton iterations, which converge quadratically
meaning that the number of iterations needed scales only doubly
logarithmically with the desired error tolerance. The square root can then be
computed via

$\sqrt{x}=\frac{x}{\sqrt{x}}.$ (822)

Thus we can use this tactic to take advantage of the rapid convergence of the
Newton iterations to the reciprocal square root by following the reversible
computation of the reciprocal square root with a single multiplication.

We represent the Fock state value (an integer) using $n$ bits. We denote $m$
to be the number of iterations needed to compute the reciprocal square root,
for example, results with $10^{-8}$ error have been obtained using $m=3$
Newton iterations [260]. In the following, the iterates will no longer be
integers, therefore, we denote $p$ the point precision and choose to set it to
$n$ such that there are as many qubits representing the number before the
floating point as there are qubits representing the number after. As in fixed-
point arithmetic, we now write the formulae with the new number of bits
$n^{\prime}=2n$.

The cost of computing a square root in terms of the number of Toffoli gates
(without un-computing the iterates) is [260]

$\begin{split}n^{\prime
2}\left(\frac{15}{2}m+3\right)+15n^{\prime}pm+n^{\prime}\left(\frac{23}{2}m+5\right)&\\\
-15p^{2}m+15pm-2m&.\end{split}$ (823)

Re-written in terms of $n$, we now have

$45n^{2}m+12n^{2}+38nm+10n-2m.$ (824)

The cost of a multiplication of two $n$-bit numbers with point precision $p$
is [260]

$N_{\rm CNOT/\times}=\frac{3}{2}n^{\prime
2}+3n^{\prime}p+\frac{3}{2}n^{\prime}-3p^{2}+3p$ (825)

which in terms of $n$ is

$9n^{2}+6n.$ (826)

If we use the standard $6$ CNOT per Toffoli gate construction [1] then the
number of CNOT gates needed to implement the square root calculation

$\ket{x}\ket{0}\mapsto\ket{x}\ket{\sqrt{x}}\ket{\psi_{\rm anc}}$ (827)

to be

$\displaystyle N_{\mathrm{CNOT/SQRT}}$
$\displaystyle=6(45n^{2}m+12n^{2}+38nm+10n-2m)+6(9n^{2}+6n)$ (828)
$\displaystyle=\left(270m+126\right)n^{2}+\left(228m+96\right)n-12m.$ (829)

If $m$ iterations are needed for the computation of the reciprocal square
root, $m$ qubit registers will be needed to store the values of the reciprocal
square root in addition to any ancillae needed in the implementation of any
binary adders that appear in the construction. Each Newton iteration requires
3 ancilla registers (which are cleaned up after each round) as well as a
register to hold the initial guess, totaling $(m+4)n^{\prime}$ qubits.
Therefore, the total number of ancillae needed to compute the square root to
$n^{\prime}$ bits of precision is

$\displaystyle n_{\textrm{qubits}}$ $\displaystyle=n^{\prime}+(m+4)n^{\prime}$
(830) $\displaystyle=2n(m+5),$ (831)

as an additional register is required to hold the initial value to carry out
the multiplication giving the square root.

Controlled rotations (CR). Unfortunately, the cost is still higher even than
this quantity. This is because controlled rotations must be employed to
implement the rotation even once such a state is prepared. That is to say if
$(\sqrt{x})_{j}$ represents the value of the $j^{\rm th}$ bit in the fixed
point representation of the square root in increasing order of significance
then

$e^{-i\sqrt{x}\sigma_{x}t}=He^{-i\sqrt{x}\sigma_{z}t}H=H\prod_{j=0}^{n-1}e^{-i(\sqrt{x})_{j}2^{j}\sigma_{z}t}H.$
(832)

Thus we can implement $e^{-iH_{\rm even}t}$ or $e^{-iH_{\rm odd}t}$ by
applying this circuit to the ancilla qubit in (815). A controlled single-qubit
rotation can be implemented using two controlled NOT gates and single-qubit
rotations. Thus the cost of the simulation of this term would also involve
another $2n$ single-qubit rotations and CNOT gates

$\displaystyle N_{\mathrm{CNOT/CR}}=2n.$ (833)

This cost is, however, negligible in most settings compared to the $O(n^{2})$
Toffoli gates needed to compute the square root.

Figure 45: Complexity of implementing a displacement operation with qubit gate
sets as opposed to bosonic gate sets after neglecting Trotter error. The
orange line represents the CNOT count for transferring the amplitude from the
$n$-qubit system register to the single main qubit, see Eq. (820). The blue
line shows the total cost of simulating the displacement operation, see Eq.
(834). The reference green dashed line is at cost 1, representing the single
displacement gate needed in bosonic hardware. This unit circuit depth metric
does not take into account the fact that the time cost of the displacement is
linear in the size of the displacement.

Cost of simulating the Displacement Operator. Combining Eq. (820), (829), and
(833), the number of two-qubit operations sufficient to implement a
displacement operation in an all-qubit device is

$\displaystyle N_{\mathrm{CNOT}}=$ $\displaystyle
N_{\mathrm{CNOT/H_{\mathrm{even}}}}+N_{\mathrm{CNOT/H_{\mathrm{odd}}}}$
$\displaystyle=$ $\displaystyle
2N_{\mathrm{CNOT/transfer}}+4N_{\mathrm{CNOT/SQRT}}+2N_{\mathrm{CNOT/CR}}$
$\displaystyle=$ $\displaystyle(270m+126)n^{2}+(144+228m)n-12m-22,$ (834)

where $n$ is the size of the qubit register required to represent the largest
initial Fock state in binary, and $m$ is the number of Newton iterations
required to calculate the inverse square root. This complexity scaling can be
seen in Fig. 45.

If we wish to represent all occupations up to a Fock state cutoff $K+1=64$,
for example, a 7-qubit register is required for the binary representation:
$n=7$. In the case where two Newton iterations $m=2$ are sufficient for our
precision goals – the level of approximation error that this corresponds to
due to Newton iterations is on the order of $10^{-4}$ [260]– this evaluates to

$\displaystyle N_{\mathrm{CNOT}}=36788.$ (835)

In contrast, the number of operations needed in the bosonic ISA is a single
(natively available) unconditional displacement operation.

The number of qubits required for a system with the largest Fock state $N$,
requiring $n$ bits for binary representation, and $m$ Newton iterations would
be

$N_{\mathrm{qubits}}=2n(m+5)+3n,$ (836)

which in this case would be 119, compared to a single oscillator plus qubit
assuming that we can neglect the cost of repeating this process via the
Trotter formula.

The Trotter error is difficult to accurately estimate as the error depends
sensitively on the input state and the signs of the commutators that arise in
a Trotter-Suzuki decomposition. In general, the number of exponentials
$N_{\exp}$ needed in a Trotter-Suzuki formula of order $2k$ to ensure that the
error is $\epsilon/2$-small is from [635]

$N_{\exp}\leq\left\lceil 3K\alpha
k\left(\frac{25}{3}\right)^{k}\left(\frac{2\alpha
K}{\epsilon}\right)^{1/2k}\right\rceil$ (837)

From Box 4.1 of Nielsen and Chuang [1] we then see that if we denote the
idealized case where the square roots are taken to be exact and take the error
in the evaluation of the square roots to be
$V_{0}(\alpha)\prod_{j=1}^{N_{\exp}/2}e^{-iH_{\rm odd}\tau_{2j-1}}e^{-iH_{\rm
even}\tau_{2j}}$ we have that the total error is bounded above by $\epsilon$
from our above error choices and the following argument.

$\displaystyle\|V(\alpha)-D(\alpha)\|$
$\displaystyle\leq\|V(\alpha)-V_{0}(\alpha)\|+\|V_{0}(\alpha)-D(\alpha)\|$
$\displaystyle\leq\frac{\epsilon}{2}+N_{\exp}\left(\frac{\alpha\epsilon}{2\alpha
N_{\exp}}\right)\leq\epsilon$ (838)

This suggests that an accurate displacement operation is likely to be even
more costly in cases where the Trotter errors come close to saturating the
worst-case estimates of (837).

Note that the above gate count does not imply that this represents a
fundamental gap between the two gate families. Instead, this shows that there
are some examples where a straightforward application of qubit compilation
techniques leads to a cost that is far greater than that needed by the bosonic
system. Regardless, the large gap between the two suggests that applications
likely exist where the costs of performing a computation in this hybrid
instruction set are lower than that of a traditional qubit-based computer.

##  Appendix G Measurements and Tomography

We have established an instruction set from which arbitrary universal
operations on the hybrid bosonic quantum processor can be achieved, which
enable useful computation and information processing tasks to be accomplished.
To extract information from the bosonic quantum processor after the
computation, measurements are needed. In this section, we shall delve more
into aspects of measurement and discuss how measurements may be performed on
these hybrid systems.

Measurements can be carried out on either qubits or oscillators, although
direct measurements on oscillators are difficult because photon number
detectors might be required. Surprisingly, by leveraging the native couplings
between the qubit and oscillator, indirect measurement of the oscillator can
be achieved by simple qubit measurement. We start by presenting a few basic
measurement schemes based on qubit measurement in Secs. G.1, G.2, G.3 to
extract photon numbers and parities. We then discuss a qubit-free homodyne
measurement scheme in Sec. G.5 that utilizes a simple oscillator-oscillator
gate. In cases where more information in the phase space of oscillators needs
to be readout, Secs. G.6, G.6, G.7, G.4 describe three different ways of
performing tomography of the phase space.

Finally, just as in DV system where random Clifford circuits combined with
measurement are used to benchmark the performance of the circuit, Gaussian
operations combined with the measurement techniques in this section can be
used to benchmark CV systems. We discuss random benchmarking of CV systems in
Sec. G.8. We note that these measurement protocols can be reformulated in the
framework of classical shadow tomography for continuous-variable systems [396,
392] and optimizations in the case of Gaussian states can also be considered
[636]. Sec. G.8 and Ref. [637] considered benchmarking of displacement
operations using a random sequence of oscillator displacements, followed by an
inverting displacement that nominally returns the system to its starting
state.

###  G.1 Hybrid Qubit-Cavity Measurements

The dispersive Hamiltonian for the coupled qubit-cavity system in Eq. (660)
commutes with both $\sigma_{z}$ and $\hat{n}=A^{\dagger}A$ and can therefore
be used for quantum non-demolition (QND) read out of both the state of the
qubit and the state of the cavity. Indeed, one of the major advantages of
storing quantum information in the large Hilbert space of an oscillator is
that state tomography is much easier and more accurate than it would be (at
least with present technology) in a qubit system with the corresponding
Hilbert space dimension.

###  G.2 Dispersive readout of the qubit

The dispersive Hamiltonian in Eq. (660) tells us that the cavity resonance
frequency is shifted to

$\omega_{R}\pm\frac{\chi}{2},$ (839)

depending on the state of the qubit. By measuring the phase shift of a
microwave pulse reflected from the cavity one can infer the resonance
frequency of the cavity and hence determine the state of the qubit (i.e.,
measure $\sigma_{z}$) with high ($\sim 99\%$) fidelity [54, 28, 638, 639].

###  G.3 Cavity photon number measurement

We can ask the question, ‘Is the cavity photon number m?’ by measuring the
value of the projector onto Fock state $|m\rangle$

$\hat{P}_{m}=|m\rangle\langle m|.$ (840)

This can be achieved in a two-step process [90] illustrated in Fig. 46a in
which we first apply the SQR gate (Box. III.9)

$U_{m}(\pi,0)=e^{i\frac{\pi}{2}\sigma_{x}\hat{P}_{m}}=(\hat{I}-\hat{P}_{m})+i\sigma_{x}\hat{P}_{m},$
(841)

which flips the qubit only if there are $m$ photons in the cavity. Then we
readout the qubit state using the resulting dispersive shift in the cavity
frequency.

Figure 46: (a) Quantum circuit using the cavity conditioned qubit rotation
gate $U_{m}(\pi/2,0)$ and a $Z$ measurement of the ancilla qubit to measure
the operator $\hat{P}_{m}$ that answers the question ‘Does the cavity contain
$m$ photons?’ (b) Circuit using $U_{n}(\pi/2,0)U_{m}(\pi/2,0)$ and a $Z$
measurement of the ancilla qubit to measure the operator
$\hat{P}_{m}+\hat{P}_{n}$ that answers the question ‘Does the cavity contain
_either_ $m$ or $n$ photons?’

Suppose that the initial cavity state is

$|\psi\rangle=\sum_{j=0}^{N_{\mathrm{max}}}\psi_{j}|j\rangle.$ (842)

If the measurement result on the ancilla qubit is $Z=+1$ the state collapses
to $|\uparrow\rangle|m\rangle$. If the measurement result on the ancilla qubit
is $Z=-1$, the state collapses to

$|\psi\rangle=|\downarrow\rangle\frac{1}{\sqrt{1-|\psi_{m}|^{2}}}\sum_{j\neq
m}\psi_{j}|j\rangle.$ (843)

These gates have been used in Sec. V.5 to present schemes that readout or
entangle dual-rail qubits using an ancilla qubit. Interestingly, we can apply
multiple versions of the cavity-conditioned bit-flip gate to ask other
questions about the cavity state. As illustrated in Fig. 46b, the compound
operator

$U_{n}(\pi,0)U_{m}(\pi,0)=e^{i\frac{\pi}{2}\sigma_{x}[\hat{P}_{m}+\hat{P}_{n}]},$
(844)

flips the qubit if the photon number is _either_ $n$ or $m$ (since it can’t be
both). These two unitaries can be executed in parallel by applying two
simultaneous drive tones on the ancilla qubit. Generalizing to a product of
many unitaries, the most general projector we can measure is

$\hat{P}_{\vec{c}}=\sum_{m=0}^{N_{\mathrm{max}}}c_{m}\hat{P}_{m},$ (845)

where $\vec{c}=(c_{0},c_{1},\ldots,c_{N_{\mathrm{max}}})$ is a binary-valued
vector, with $c_{j}\in\\{0,1\\};j=0,N_{\mathrm{max}}$.

This is a powerful gate. One could use it for example to flip the ancilla bit
if the photon number is less than (or greater than) a selected value. One
could then use the qubit-conditioned cavity displacement gate (Box. III.7) to
displace the cavity conditioned on the photon number being less than (or
greater than) the selected value. As we describe further below, it can also be
used in a binary search for the photon number.

####  G.3.1 Cavity Photon Number Parity Measurement

A particularly interesting and useful example of the general projectors
available in the class $\hat{P}_{\vec{c}}$ is the photon number parity
operator

$\hat{\Pi}=e^{i\pi\hat{n}}=\sum_{m\in\mathrm{odd}}\hat{P}_{m},$ (846)

which utilizes $\vec{c}=(0,1,0,1,0,1,0\ldots)$. The photon parity operator
plays an essential role in cavity state tomography (to be described below) and
in quantum error correction using bosonic codes [48]. The actual pulse
sequence used in practice to measure the parity [130] is different than that
outlined above but produces the same result (see Sec. V.5). The circuit used
takes advantage of the conditional cavity rotation afforded directly under
time evolution with the dispersive Hamiltonian in Eq. (660). Time evolution
under the dispersive coupling

$\frac{\chi}{2}\sigma_{z}A^{\dagger}A$ (847)

for a time $t=-\theta/\chi$ ($\chi$ is typically negative) is described by the
‘conditional rotation’ unitary (Box III.8)

$\mathrm{CR}(\theta)=e^{-i\frac{\theta}{2}\sigma_{z}\hat{n}}.$ (848)

Applying the phase estimation protocol to this unitary will give us a powerful
way to measure the full photon number distribution along with photon number
parity and ‘super-parity’ measurements [90].

To see the connection to the parity operator, consider the conditional
rotation gate for $\theta=\pi$ (Box III.8)

$\mathrm{CR}\left(\pi\right)=e^{-i\frac{\pi}{2}\hat{n}}\,\mathrm{CP},$ (849)

where

$\mathrm{CP}=e^{i\pi|e\rangle\langle e|\hat{n}}=|g\rangle\langle
g|\hat{I}+|e\rangle\langle e|\hat{\Pi}$ (850)

is the qubit-controlled cavity parity operation which is active only if the
qubit is in its excited state $|e\rangle$. The additional rotation of the
cavity state in Eq. (849) corresponds for our purposes to an irrelevant frame
choice. The circuit shown in Fig. 47, familiar with the phase estimation
problem, takes advantage of the phase kickback on the ancilla from the
controlled parity unitary to entangle the qubit state with the cavity parity,
thus enabling the cavity parity to be measured by reading out the qubit state.
The measurement result $z_{0}=\pm 1$ determines the parity bit of the photon
number via

$b_{0}=\frac{1-z_{0}}{2}$ (851)

Figure 47: Quantum circuit using phase kickback from the controlled oscillator
rotation, ${\rm CR}(\pi)$, to entangle the ancilla qubit state with the photon
number parity of the mode. Measurement of the qubit state then yields a QND
measurement of the parity of the photon number.

###  G.4 Cavity Photon Number ‘Super-Parity’ Measurement

We can use phase estimation to generalize the above result to obtain the
‘super-parities’ $b_{j}$ that yield the full binary number representation of
the photon number

$n=\sum_{j=0}^{j_{\mathrm{max}}}2^{j}b_{j}.$ (852)

If the measurement result for $b_{0}$ is known, then we can determine the next
binary digit by measuring $b_{1}=\frac{[(n-b_{0})\,\mathrm{mod}\,4]}{2}$. This
requires the circuit in Fig. 47 by modifying the ${\rm CR(\pi)}$ gate into the
following unitary

$\mathrm{CR}\left(\frac{\pi}{2}\right)R_{z}\left(-b_{0}\frac{\pi}{2}\right),$
(853)

where the $R_{z}$ is the single-qubit rotation in Table A.1 (again executable
‘in software’ via a frame change)

$R_{z}(\theta)=e^{-i\frac{\theta}{2}\sigma_{z}}.$ (854)

It can be shown that

$\displaystyle b_{1}=\frac{1-z_{1}}{2}$ (855)

where $z_{1}$ is the measurement outcome of the qubit.

Similarly, once the measurement results for $b_{0}$ and $b_{1}$ are known,
$b_{2}$ is obtained using the unitary

${\rm
CR}\left(\frac{\pi}{4}\right)R_{z}\left(-[b_{0}+2b_{1}]\frac{\pi}{4}\right),$
(856)

and $b_{3}$ is obtained using the unitary

${\rm
CR}\left(\frac{\pi}{8}\right)R_{z}\left(-[b_{0}+2b_{1}+4b_{2}]\frac{\pi}{8}\right).$
(857)

In general, $b_{k}$ can be obtained using the unitary

$\displaystyle{\rm
CR}\left(\frac{\pi}{2^{k}}\right)R_{z}\left(-\left[\sum_{j=0}^{k-1}2^{j}b_{j}\right]\frac{\pi}{2^{k}}\right).$
(858)

Importantly, since all these unitaries commute with each other we can make a
QND measurement of all the binary digits in a single shot (single state
preparation) using the quantum circuit with classical feed-forward shown in
Fig. 48. This is very powerful because it effectively performs a binary search
to obtain the value of $n$ in only $\sim\log N_{\mathrm{max}}$ steps. It is
exponentially more efficient than using the circuit in Fig. 46a to
sequentially ask, ‘Is $n$ equal to zero?, Is $n$ equal to 1? Is $n$ equal to
two? …’ and so on.

Figure 48: Quantum circuit with classical feed-forward of the ancilla
measurement results $b_{0},b_{1},\ldots$ to carry out the phase estimation
algorithm to measure the binary representation $(\ldots,b_{1},b_{0})$ of the
photon number.

As an alternative to the superparity measurements which determines the least-
significant bit (LSB) first, we can perform a more traditional binary search
for the photon number value by starting from the most-significant bit (MSB).
If we consider a range of photon numbers $[0,N_{\mathrm{max}}]$, with
$N_{\mathrm{max}}+1$ being an integer power of 2, then we could use the
projector in Eq. (845) to ask, ‘Is the photon number in the upper half of the
range? (i.e. in the interval $[(N_{\mathrm{max}}+1)/2,N_{\mathrm{max}}]$).
Given the answer, we then recursively ask,‘ Is the photon number in the upper
half of the interval containing the photon number?’ This is just like
classical binary search. In the spirit of Eqs. (844,845), let us define a
generalized CNOT gate controlled by the state of the cavity

$U_{\vec{c}}\,(\pi)=e^{-i\frac{\pi}{2}\sigma_{x}\hat{P}_{\vec{c}}}.$ (859)

Now consider the use of this gate in the circuit depicted in Fig. 49. Note
that in analyzing this sequence of measurements, we will assume that auxiliary
qubit is initialized in state $|0\rangle$ at the beginning of the protocol,
but is _not_ reset after each measurement.

Figure 49: Quantum circuit for binary search to measure photon number
$\hat{n}$. Unlike the phase estimation protocol, this does not require with
classical feed-forward of the ancilla measurement results
$a_{0},a_{1},a_{2},\ldots$. The binary representation of the photon number can
be determined directly from the sequence of measurement results. Note that the
auxiliary qubit is assumed to be initialized in $|g\rangle=|0\rangle$, but is
not reset after each measurement. Ref. [90] uses essentially this protocol but
_does_ reset the qubit after each measurement.

In this simple example, we take the maximum photon number to be
$N_{\mathrm{max}}=7$ and the vector $\vec{c}_{j}$ to be the $j$th basis vector
of the Walsh-Hadamard transform:

$\displaystyle\vec{c}_{2}$ $\displaystyle=$ $\displaystyle(00001111)$ (860)
$\displaystyle\vec{c}_{1}$ $\displaystyle=$ $\displaystyle(00110011)$ (861)
$\displaystyle\vec{c}_{0}$ $\displaystyle=$ $\displaystyle(01010101).$ (862)

From the vectors $\\{\vec{c}_{2},\vec{c}_{1},\vec{c}_{0}\\}$ we see that the
measurement result $a_{2}$ tells us whether the photon number is in the upper
half $[4,7]$ ($a_{2}=1,\ket{e}_{\rm qubit}$) or the lower half $[0,3]$
($a_{2}=0,\ket{g}_{\rm qubit}$) of the considered range of $[0,7]$. If the
measurement result in $a_{1}$ is different than $a_{2}$, the ancilla spin has
flipped and therefore the photon number is in the upper half of the relevant
range. Similarly, if the measurement result $a_{0}$ is different than $a_{1}$
the spin has again flipped and the photon number parity must be odd. Thus we
obtain the binary digits representing the photon number

$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle a_{2},$ (863)
$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle a_{2}\oplus a_{1}$ (864)
$\displaystyle b_{0}$ $\displaystyle=$ $\displaystyle a_{1}\oplus a_{0},$
(865)

where $\oplus$ corresponds to addition modulo 2.

The fact that active feed-forward is not required is also a significant
advantage. This is (essentially) the technique used in the boson sampling
experiment of Wang et al. [90], except those authors used numerically
generated optimal control pulses on the qubit to map the binary bits in the
photon number onto an ancilla. The other difference is that Wang et al. reset
the ancilla to its ground state after each measurement to minimize errors from
the energy relaxation of the ancilla.

###  G.5 Homodyne Detection

Homodyne detection can be seen as a projective measurement of the phase-space
quadratures. The setup for homodyne detection includes a beam splitter with
transmission coefficient (l) and reflection coefficient (r), and two
photodetectors (see figure 50). The quantity of interest here is

$\langle n_{3}\rangle-\langle n_{4}\rangle$ (866)

the difference of mean photon numbers detected by the two detectors for any
input state.

Figure 50: Homodyne detection involves a beam splitter (BS) with transmission
coefficient ‘t’ and reflection coefficient ‘r’, and two detectors at the
output ports of the BS. For balanced homodyne detection,
$|t|^{2}=|r|^{2}=\frac{1}{2}$.

The state to be measured ($\ket{\psi}$) is sent through one port while the
coherent state ($\ket{\alpha}$) is sent through another port. In the
Heisenberg picture, the photon number operators at the output ports is,

$\displaystyle n_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[|t|^{2}n_{1}+|r|^{2}n_{2}-i(t^{*}ra_{1}^{\dagger}a_{2}-r^{*}ta_{2}^{\dagger}a_{1})\right]$
$\displaystyle n_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[|t|^{2}n_{2}+|r|^{2}n_{1}-i(t^{*}ra_{2}^{\dagger}a_{1}-r^{*}ta_{1}^{\dagger}a_{2})\right]\,,$

and the mean photon number is given by

$\displaystyle\langle n\rangle$ $\displaystyle=$
$\displaystyle\bra{\psi,\alpha}n\ket{\psi,\alpha},$ (869)
$\displaystyle\langle n_{3}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\rule{0.0pt}{10.33327pt}|t|^{2}\bra{\psi}n_{1}\ket{\psi}+|r|^{2}|\alpha|^{2}\right.~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}\left.-i\bra{\psi}(tr^{*}\alpha)a_{1}^{\dagger}-(tr^{*}\alpha)^{*}a_{1}\ket{\psi}\rule{0.0pt}{10.33327pt}\right].$

Assuming a balanced homodyne detection with $t=r$ and
$\alpha=|\alpha|e^{i\phi}$

$\langle n_{3}\rangle-\langle
n_{4}\rangle=|\alpha|\bra{\psi}(a^{\dagger}e^{i\phi}-ae^{-i\phi})\ket{\psi}.$
(871)

Thus, the mean photon number measures the mean value of a phase-space
quadrature along an axis dependent on the phase of the coherent state. This is
a multi-purpose method. Homodyne detection on identically prepared states
reconstructs the quadrature probability distribution. Repeating these
measurements for all different phases is equivalent to computing the marginal
integrals of Wigner function, an important visualization tool for quantum
states, which then can be used to obtain the Wigner function (and the density
operator) via a Radon transform [640, 641, 642, 643].

Homodyne detection of traveling photons at optical frequencies can be
relatively efficient, but at microwave frequencies, even nearly quantum-
limited linear amplifiers add enough noise to complicate state reconstruction,
but with sufficient care, this can be done relatively well [532]. For
microwave photons trapped in stationary modes in high $Q$ resonators, it is
possible to avoid these difficulties by synthesizing homodyne and heterodyne
measurements by mapping the cavity state onto an auxiliary qubit and measuring
the qubit [296].

###  G.6 Wigner and Characteristic Function Tomography

A key step in validating quantum processes is to estimate the Wigner function
for a quantum state. This allows us to assess whether there are errors in our
quantum circuits. Here we review processes for estimating the characteristic
function, the Wigner function, and the Husimi $Q$-function.

The phase-space density functions $W$ (Wigner), $Q$, and $P$ are different
Fourier transforms of characteristic functions. A characteristic function is
the average of the displacement operator at each point in phase space for a
given density matrix $\rho$,

$C(\alpha)=\mathrm{Tr}\left[\rule{0.0pt}{10.33327pt}\rho D(\alpha)\right],$
(872)

where $\mathcal{D}(\alpha)$ is the (unconditional) displacement operator
defined in Box III.1. It can be seen that for a pure state, this quantity is
an auto-correlation function of the quantum state in phase space. $C(\alpha)$
is often referred to as the symmetric characteristic function to distinguish
it from its normal ($C_{n}$) and anti-normal ($C_{an}$) order counterparts,

$\displaystyle C_{n}(\alpha)=\mathrm{Tr}[\rho e^{\alpha
a^{\dagger}}e^{-\alpha^{*}a}],$ (873) $\displaystyle
C_{an}(\alpha)=\mathrm{Tr}[\rho e^{-\alpha^{*}a}e^{\alpha a^{\dagger}}].$
(874)

The Fourier transform of the symmetric characteristic function $C(\alpha)$ is
called the Wigner function and is defined in Eqs. (364-365) in the main body
of the text and is restated here for convenience:

$W(\beta)=\frac{1}{\pi^{2}}\int d^{2}\alpha\
C(\alpha)e^{\beta\alpha^{*}-\alpha\beta^{*}}.$ (875)

The density matrix $\rho$ of a system contains the full information needed to
predict the (statistical) outcomes of any measurement on that system. The
Wigner function is a quasi-probability distribution in phase
(position/momentum) space that contains the equivalent information to the
density matrix but is easier to measure. Moreover, practical methods for
Wigner function estimation have been developed in the literature [644].

In the same dimensionless complex coordinates that we use to describe the
amplitude of a coherent state, the relationship of the Wigner function to the
density matrix is given by the following remarkable identity due to Lutterbach
and Davidovich, [92]

$W(\beta)=\frac{2}{\pi}\mathrm{Tr}\left[\rho{\mathcal{D}}^{\dagger}(-\beta)\
\hat{\Pi}\ {\mathcal{D}}(-\beta)\right]\,.$ (876)

where $\hat{\Pi}$ is the photon number parity operator as defined in Eq.
(846).

This formula tells us that the value of the Wigner function at the point
$\beta$ in the complex plane is determined by displacing that point to the
origin and then measuring the expectation value of the photon number parity.
This remarkable result tells us that state tomography on the oscillator is
relatively easy to obtain, even though the effective dimension of the Hilbert
space $N_{\mathrm{max}}+1$ can be quite large (though see the following
paragraph for an important caveat regarding sampling complexity). In addition,
the basic procedure does not change as $N_{\mathrm{max}}$ increases. It is
worth noting that Eq. (876) shows that determining only the photon number
parity, instead of the full photon number distribution, suffices for Wigner
function tomography. However, Ref. [645] demonstrated that the redundant
information available in the full photon number distribution can significantly
improve the robustness of state tomography via the Wigner and characteristic
functions.

Despite the simplicity of determining $W(\beta)$ for a single point in phase
space, the cost of performing full Wigner function characterization using the
above relation faces challenges if we wish to learn the entire Wigner function
of even a single oscillator. The reason for this is that an infinite number of
values of $\beta$ are needed to ensure that the Wigner function can be
accurately estimated at any $\beta$. This means that we will ultimately be
forced to estimate the value of the Wigner function at a given point by using
a scheme such as a Lagrange interpolation, wherein the value of the Wigner
function at an arbitrary point is estimated from values learned at points on a
grid. Assuming the Wigner function is differentiable $n+1$ times, from
interpolation theory (in essence Taylor’s remainder theorem), the error in
estimating $W(\beta)$ at a particular point $\beta$ from surrounding
gridpoints with a polynomial function $P_{n}(\beta)$ with constant grid
spacing $h$ is

$|W(\beta)-P_{n}(\beta)|\leq\frac{\max_{y,\alpha}|D^{n+1}_{\alpha}W(y)|h^{n}}{(n+1)!},$
(877)

where $D_{\alpha}$ is the directional derivative operator in the direction
$\alpha$. This suggests that if we wish to estimate the Wigner function at an
arbitrary point within error $\epsilon$ for an unknown Wigner function, that
is differentiable at least $n+1$ times then it suffices to choose

$h\in
O\left(\frac{n\epsilon^{1/n}}{\max_{y,\alpha}|D_{\alpha}^{n+1}W(y)|^{1/n}}\right).$
(878)

If we assume that the Wigner function is (within sufficiently small error)
supported on a compact region $[-\beta_{\max},\beta_{\max}]^{\otimes 2}$, then
since the Wigner function for a single oscillator is a two-dimensional
function, the total number of grid points needed within this space scales as

$N_{1}=\left(\frac{\beta_{\max}}{h}\right)^{2}\subseteq
O\left(\left(\frac{\beta_{\max}\max_{y,\alpha}|D_{\alpha}^{n+1}W(y)|^{1/n}}{n\epsilon^{1/n}}\right)^{2}\right)$
(879)

Repeating the same analysis for a system with $p\geq 1$ oscillators (but
taking $y,\beta,\alpha$ now to be vectors in a $2p-$dimensional space), we
conclude that the number of points needed to extrapolate the Wigner
distribution is

$\displaystyle N_{p}$ $\displaystyle=\left(\frac{\beta_{\max}}{h}\right)^{2p}$
$\displaystyle\subseteq
O\left(\left(\frac{p\beta_{\max}\max_{y,\alpha}|D_{\alpha}^{n+1}W(y)|^{1/n}}{n\epsilon^{1/n}}\right)^{2p}\right)$
(880)

Choosing for simplicity $n=2p$, we see that the scaling is

$\displaystyle N_{p}$
$\displaystyle=O\left(\frac{\max_{y,\alpha}|D_{\alpha}^{2p+1}W(y)|}{\epsilon}\left(\frac{\beta_{\max}}{2}\right)^{2p}\right)$
(881)

We see from this that unless the derivatives of the Wigner function become
very small as $p$ increases, the number of points that need to be accurately
estimated to approximately learn the Wigner function on $p$ oscillators scales
exponentially with $p$. This can be problematic even for physical states. For
example, if we consider a coherent state in one dimension of the form

$\ket{\psi}^{\otimes j}=\left(\int dq\
e^{-q^{2}/2\sigma^{2}}\ket{q}/\sqrt{2\pi}\sigma\right)^{\otimes j}$ (882)

then for small $\sigma$ we have that the norm of the derivatives of the state
grows exponentially with the number of copies $j$. This illustrates the often-
observed fact with conventional tomography that the cost grows exponentially
with the number of systems.

This exponential increase of the number of measurement settings with the
number of qumodes in the system exactly matches what we would expect from
qubit tomography of a density matrix (with no promises about the rank of the
density operator) that $4^{p}$ distinct sets of measurements of Pauli
operators would be needed to reconstruct a $p$-qubit state [646]. This
suggests that the situation is analogous, if more complex, to the qubit case
as we have to approximately learn the density matrix because we do not have a
tomographically complete set of measurements for our continuous scheme. Thus
while individual components of the Wigner function can be easily estimated,
learning the overall Wigner function for a generic $p$-mode oscillator system
will likely be forever out of reach owing to the previous argument about the
scaling with the derivatives of the tensor products between coherent states.
However, for cases where prior knowledge is provided about the form of the
Wigner function, more efficient reconstruction schemes can be found and the
optimization of such reconstruction schemes remains an important open
question.

###  G.7 The Husimi Q-Function

The Fourier transform of the anti-normal order characteristic function
$C_{an}(\alpha)$ in Eq. (874) is called the Husimi-$Q$ function, and is
defined as

$Q(\beta)=\frac{1}{\pi^{2}}\int d^{2}\alpha\
C_{an}(\alpha)e^{\beta\alpha^{*}-\alpha\beta^{*}}\,.$ (883)

The Husimi-$Q$ function can be more convenient to use in certain cases since
unlike $W$, this quantity is always positive while it is less-suited to study
the non-classicality of quantum states. In terms of the density matrix, it is
given by

$Q(\beta)=\frac{1}{\pi}\mathrm{Tr}\left[\rule{0.0pt}{10.33327pt}\ket{0}\bra{0}{\mathcal{D}}^{\dagger}(-\beta)\
\rho\ {\mathcal{D}}(-\beta)\right].$ (884)

Thus, it is the average of the projector onto the vacuum state, of the state
displaced by $\beta$.

Similar to the definition of the $Q$-function, the $P$-function is the Fourier
transform of $C_{n}(\alpha)$ in Eq. (873). It is unnormalizable and difficult
to use and hence will not be discussed in this document.

###  G.8 Calibrating Controlled Displacements.

As discussed in Sec. IV.5.2, benchmarking hybrid quantum processors is not as
easy as qubit systems. However, some simple calibrations on hybrid CV/DV gates
can still be performed. One example is that the gate sequence used in Eq.
(413) in Sec. V can be used to calibrate the controlled displacement gate.

Figure 51: Average measurement results for the qubit measurement outcome using
the calibration given in Eq. (885), demonstrating ‘Berry fringe’ oscillations
due to the phase kickback of the non-commuting displacements in phase space.
The known form of these oscillations from Eq. (886) can be used to accurately
calibrate the size of displacements in phase space.

It works as follows. Consider conjugating Eq. (413) using Hadamard gates to
construct the following $U_{\rm calibration}$

$\displaystyle U_{\rm
calibration}=H\left[\rule{0.0pt}{10.33327pt}D(-i\alpha)D_{c}(\alpha,-\alpha)D(i\alpha)D_{c}(-\alpha,\alpha)\right]H.$
(885)

As we have seen from Eq. (414) and Fig. 18, the $|0\rangle$ state of the qubit
causes the displacements to drive the oscillator around the counterclockwise
closed loop in phase space, which means that the $|0\rangle$ state acquires
phase $\varphi_{|0\rangle}=-2|\alpha|^{2}$. Similarly, the $|1\rangle$ state
of the qubit causes the displacements to drive the oscillator around the
clockwise closed loop in phase space, which means that the $|1\rangle$ state
acquires phase $\varphi_{|1\rangle}=+2|\alpha|^{2}$. The net effect is that
the qubit undergoes a rotation around the $z$ axis through an angle
$\varphi=\varphi_{|0\rangle}-\varphi_{|1\rangle}=-4|\alpha|^{2}$. The
conjugation by the Hadamard gates converts this to a rotation around the $x$
axis leading to the ‘Berry fringe’ oscillations

$\langle Z\rangle=\cos\left(4\alpha^{2}\right),$ (886)

as is shown in Fig. 51. Therefore, by accumulating enough shots on the qubit
measurement, we can get an estimation of $\langle Z\rangle$ to infer the
actual displacement amount $\alpha$ performed by the displacement gates. Note
that if we make the unconditional displacements in Eq. (885) be constant $\pm
i\beta$ independent of $\alpha$ then we obtain a simple sinusoidal oscillation

$\langle Z\rangle=\cos\left(4\alpha\beta\right).$ (887)

Ref. [637] takes a different approach, considering a set of random
displacements applied to an initial vacuum state. This is followed by an
inverting displacement that nominally returns the system to the vacuum state.
Measurement of the probability of being in the vacuum state (or more
generally, measurement of the photon number distribution) provides information
on noise and errors in the displacement operations.

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