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# DQC1 as an Open Quantum System

Jake Xuereb  Department of Physics, University of Malta, Msida MSD 2080, Malta
<EMAIL_ADDRESS>School of Physics, University College Dublin,
Belfield, Dublin 4, Ireland  School of Physics, Trinity College Dublin,
College Green, Dublin 2, Ireland  Steve Campbell  School of Physics,
University College Dublin, Belfield, Dublin 4, Ireland  Centre for Quantum
Engineering, Science, and Technology, University College Dublin, Belfield,
Dublin 4, Ireland  John Goold  School of Physics, Trinity College Dublin,
College Green, Dublin 2, Ireland  André Xuereb  Department of Physics,
University of Malta, Msida MSD 2080, Malta

###### Abstract

The DQC1 complexity class, or power of one qubit model, is examined as an open
quantum system. We study the dynamics of a register of qubits carrying out a
DQC1 algorithm and show that, for any algorithm in the complexity class, the
evolution of the logical qubit can be described as an open quantum system
undergoing a dynamics which is unital. Unital quantum channels respect the
Tasaki-Crooks fluctuation theorem and we demonstrate how this is captured by
the thermodynamics of the logical qubit. As an application, we investigate the
equilibrium and non-equilibrium thermodynamics of the DQC1 trace estimation
algorithm. We show that different computational inputs, i.e. different traces
being estimated, lead to different energetic exchanges across the register of
qubits and that the temperature of the logical qubit impacts the magnitude of
fluctuations experienced and quality of the algorithm.

Figure 1: The main ideas of this work visualised, (a) a quantum circuit can be
thought of as (b) an effective open quantum system whose dynamics be analysed
through the formalism of (c) quantum channels.

##  1 Introduction

A quantum algorithm in gate based quantum computation is a process where the
information content of a quantum state is changed towards achieving a
computational goal upon measurement. Since information and thermodynamics can,
in certain contexts, be two sides of the same coin one would expect that
relationships can be made between a computational task and the thermodynamics
experienced by the system carrying out an algorithm to achieve this goal.

Large strides have been made towards understanding the relationship between
quantum information and thermodynamics [1, 2, 3, 4, 5, 6, 7, 8] but the
thermodynamics of quantum computation itself, as is highlighted in [9], has
been largely unexamined despite its clear importance. Recent work has started
to move in this direction with the study of thermodynamics of quantum
annealers [10, 11, 12], investigations of heat engines in quantum computation
[13, 14] and thermodynamic optimisation of information processing tasks
focusing on changes in numbers of qubits [15, 7]. But an analysis of the
thermodynamics of a quantum system in light of the content of the computation
it is carrying out has not been performed, to our knowledge.

In this work we attempt to address this gap by making use of the formalism of
quantum channels to model arguably the simplest quantum complexity class as an
open quantum system. After introducing the DQC1 complexity class in Section 2,
we prove our main result in Section 3 by showing that the dynamics of the
logical qubit of any DQC1 algorithm is unital [16, 17] in the sense of open
quantum systems. In Section 4 we discuss how this implies that these dynamics
satisfy the Tasaki-Crooks fluctuation theorem [18, 19, 20, 21] and in Section
5 present the thermodynamics of the trace estimation algorithm as an
application. Here we find that different trace estimations lead to different
energetic exchanges amongst the qubits of the DQC1 register and that the
temperature of the control qubit in this algorithm impacts the amount of
fluctuations one expects in this model.

##  2 The DQC1 Complexity Class

The deterministic quantum computation with one-clean-qubit model (DQC1) is a
restricted class of quantum algorithms satisfying the following properties:

  1. 1.

the initial state is $\rho=|0\rangle\\!\langle 0|\otimes I_{n}/2^{n}$, that is
a clean qubit in a pure state and an $n$-qubit ancillary register in the
maximally mixed state;

  2. 2.

a poly($n$)-size gate can be enacted across the $n+1$ qubits;

  3. 3.

the clean qubit is measured giving the output bit $a\in\\{0,1\\}$.

Introduced by Knill & Laflamme to investigate the power of one bit of quantum
information [22], DQC1 is particularly suited to NMR architectures [22, 23],
and has been studied as a complexity class in its own right [24, 25, 26, 27].
It is capable of solving various problems, [28, 24, 29, 30] some of which are
classically hard and has also been studied as a potential quantum machine
learning model [31]. In particular, the trace estimation of unitary matrices
has been shown to be DQC1-complete [24]. All this despite being incapable of
universal quantum computation [32].

The initial state of DQC1 circuits may be expressed as

$\displaystyle\rho=|0\rangle\\!\langle
0|\otimes\frac{I_{n}}{2^{n}}=\frac{I_{1}\otimes I_{n}+\sigma^{z}\otimes
I_{n}}{2^{n+1}}=\frac{I_{n+1}+\sigma^{z}_{1}}{2^{n+1}}$ (1)

where $\sigma^{z}$ acts on the first qubit. This means that the state at the
end is given by

$\displaystyle\rho^{\prime}=\frac{I_{n+1}+V\sigma^{z}_{1}V^{\dagger}}{2^{n+1}}$
(2)

where $V$ is the product of unitaries implemented across the DQC1 register
during the computation. If one then defines [24]

$\displaystyle\mu=\frac{\text{tr}\\{V\sigma^{z}_{1}V^{\dagger}\sigma^{z}_{1}\\}}{2^{n+1}}$
$\displaystyle
R=\frac{V\sigma^{z}_{1}V^{\dagger}-\mu\sigma^{z}_{1}}{\sqrt{1-\mu^{2}}}$

the final state before measurement can be expressed as

$\displaystyle\rho^{\prime}=\frac{I_{n+1}+\mu\sigma^{z}_{1}+\sqrt{1-\mu^{2}}R}{2^{n+1}}$
(3)

where $R$ and $\sigma^{z}_{1}$ are traceless and $\mu$ is real. The
probability of measuring $0$ on the first qubit is then

$\displaystyle\mathcal{P}[0]=\text{tr}\left\\{\rho^{\prime}\frac{I_{1}+\sigma^{z}_{1}}{2}\right\\}=\frac{1+\mu}{2}.$

Applying this structure more specifically to estimate the trace of some $U$
using the circuit in Figure 1 (a) we have

$\displaystyle\rho^{\prime}=\frac{I_{n+1}+\sigma^{z}_{1}\otimes\left(\frac{U+U^{\dagger}}{2}\right)+\sigma^{y}_{1}\otimes\left(\frac{U-U^{\dagger}}{2i}\right)}{2^{n+1}}$
(4)

and therefore measuring with respect to $\sigma^{z}_{1}$ gives
$\mu=\frac{\Re\\{\text{tr}U\\}}{2^{n}}$.

The work of Linden & Jozsa [33] demonstrated that quantum computation
involving pure states necessarily requires the establishment of multipartite
entanglement. However, the situation is less clear cut when dealing with mixed
states as in DQC1, particularly when we allow for the logical qubit to also be
in a mixed state. We can consider a generalisation of the DQC1 trace
estimation algorithm when the first qubit is not clean but possesses some
polarisation $\alpha$

$\displaystyle\rho(\alpha)=\frac{I_{1}+\alpha\sigma^{z}}{2}\otimes\frac{I_{n}}{2^{n}}$
$\displaystyle\rho^{\prime}(\alpha)=\frac{I_{n+1}+\alpha\sigma^{z}_{1}\otimes\left(\frac{U+U^{\dagger}}{2}\right)+\alpha\sigma^{y}_{1}\otimes\left(\frac{U-U^{\dagger}}{2i}\right)}{2^{n+1}}$

as introduced in [34] as a prototypical circuit to study the resource giving
advantage is this setting. Here it has been shown that no distillible
entanglement, quantified by negativity, is formed across the first qubit and
the fully-mixed qubits but that non-negligible amounts of negativity are found
in bipartitions of the fully mixed qubits and the first qubit. In later work
[35], a measure of non-classical correlations called the quantum discord [36,
37] was shown to be non-zero between the first qubit and the mixed qubits
leading to the argument that the quantum discord is the resource behind the
speed-up in trace estimation offered by DQC1. This claim has been argued
against in [38] by counterexample and more recently in a very robust argument
made by Cade and Yoganathan who showed that the DQC1 trace estimation circuit
is classically simulable for the class of unitaries that do not generate
entanglement, that is DQC1${}_{sep}\subseteq$ BPP [30].

This generalisation is referred to as the one-non-clean-qubit model and was
studied in [39] as a computational model. It is more representative of what is
physically achievable with NMR since the clean qubit will always have some
degree of polarisation. As such, knowing how this polarisation imparts error
in the computation is relevant.

##  3 Unitality of DQC1 Channels

Having introduced the DQC1 complexity class, we now demonstrate that the
dynamics of the logical can be recast as a quantum channel using the tools of
open quantum systems, illustrated in Fig 1 (b,c).

In this formalism we may associate this evolution with a Stinespring dilated
representation as

$\displaystyle\mathcal{E}(\rho_{S})=\text{tr}_{n}\left\\{V\left(\rho_{S}\otimes\frac{I_{n}}{2}\right)V^{\dagger}\right\\}=\rho^{\prime}_{S}$
(5)

where $V$ is a unitary satisfying property 2 representing the product of
unitary gates carried out on the register which result in a specific DQC1
algorithm. By the Choi-Kraus Theorem we can associate a Kraus representation
to this map

$\displaystyle\mathcal{E}(\rho_{S})=\sum_{i}\Lambda_{i}\rho_{S}\Lambda^{\dagger}_{i}\,:\,\sum_{i}\Lambda_{i}^{\dagger}\Lambda_{i}=I_{1}$
(6)

where $\Lambda_{i}$ are the corresponding Kraus operators which are derived
from the unitary $V$ in the Stinespring dilation as

$\displaystyle\Lambda_{i,j}=\sqrt{b_{j}}\bra{i_{B}}V\ket{j_{B}}$ (7)

where $\ket{i_{B}}$, $\ket{j_{B}}$ are eigenkets of the environment and
$b_{j}$ are the eigenvalues of the initial state of the environment. Note that
we are using two indices since the environment is in a mixed state. In the
DQC1 setting we assume the ancillary register is in the maximally mixed state
and plays the role of the environment. The state of our environment is
diagonal in the energy eigenbasis of a local Hamiltonian
$H=\sum_{i}E_{i}|E_{i}\rangle\\!\langle E_{i}|$ since it can be expressed
$\frac{I_{n}}{2^{n}}=\frac{1}{2^{n}}\sum_{i}|E_{i}\rangle\\!\langle E_{i}|$.
As such, the Kraus operators $\Lambda_{i}$ can be expressed directly as

$\displaystyle\Lambda_{i,j}=\frac{1}{\sqrt{2^{n}}}\bra{E_{i}}V\ket{E_{j}}$ (8)

and considering

$\displaystyle\sum^{2^{n}}_{i,j=1}\Lambda_{i,j}\Lambda^{\dagger}_{i,j}$
$\displaystyle=\sum^{2^{n}}_{i,j=1}\frac{1}{2^{n}}\bra{E_{i}}V|E_{j}\rangle\\!\langle
E_{j}|V^{\dagger}\ket{E_{i}}$ $\displaystyle=I_{1}$

by completeness of the energy eigenbasis and unitarity of $V$, we have that
$\mathcal{E}\left(\cdot\right)$ is a unital channel for any DQC1 algorithm.
This is our first result.

To understand the origin of unitality we can consider instead the register to
be in an arbitrary mixed state giving

$\displaystyle\sum^{2^{n}}_{i,j=1}\Lambda_{i,j}\Lambda^{\dagger}_{i,j}=\sum^{2^{n}}_{i,j=1}\sqrt{b_{j}}\bra{E_{i}}V|E_{j}\rangle\\!\langle
E_{j}|V^{\dagger}\ket{E_{i}}$ (9)

where now we are unable to use the completeness relation unless $b_{j}$ is
either constant, as in the case of a maximally mixed state, or has only a
single non-zero contribution, as in the case of the ancillary qubits being in
the ground state. The unitality of DQC1 channels thus emerges as a result of
the environment, i.e. the ancillary register found in the maximally mixed
state and property 1 of this complexity class. We provide some further details
regarding the origin of unitality in an algorithmic setting in Appendix A.

To summarise, DQC1 algorithms encode all relevant computational information in
the state of the logical qubit and this is the only qubit which is measured as
described by property 3. In light of this, it is relevant to examine the
evolution of this qubit throughout this quantum information process. We have
shown that the dynamics of this evolution is unital. In the proceeding
sections, we will apply this fact to the examine the thermodynamics and
energetics of this complexity class and specifically of the trace estimation
algorithm.

##  4 DQC1 & the Tasaki-Crooks Fluctuation Theorem

Quantum channels [40, 17, 41] provide an operational formalism to describe
open quantum system dynamics without the direct use of a master equation [42].
As such, they carry information about the thermodynamics of the process they
describe. In particular, it has been shown that unital channels [16, 17] model
processes that satisfy the Tasaki-Crooks fluctuation theorem [43, 44, 45]
which relates the ratio of the forward and reverse work probability
distributions to the entropy produced by the process. Therefore, as a
corollary of the result from Section 3, we have that the dynamics of the
logical qubit in any DQC1 algorithm necessarily respects the Tasaki-Crooks
Theorem. In this section we explore this corollary and its implications in the
context of DQC1 trace estimation with a non-clean qubit as examined in [34,
35, 46] to gain an operational understanding of what the Fluctuation theorem
could mean for a quantum algorithm.

Consider a DQC1-channel Eq. (5) with $V=(H\otimes
I_{n})\left(|0\rangle\\!\langle 0|\otimes I_{n}+|1\rangle\\!\langle 1|\otimes
U\right)(H\otimes I_{n})$ where $H$ is the Hadamard gate on the logical qubit,
$I_{n}$ is the identity operation on the $n$-qubit ancillary register and $U$
is the unitary which is applied to the ancillary register conditioned on the
logical qubit. It is the trace of this $U$ which we estimate by measuring the
logical qubit.

The logical qubit in this algorithm is treated as the energetic system of
interest, with initial state

$\displaystyle\rho_{C}(\alpha)=\frac{I_{1}+\alpha\sigma^{z}_{1}}{2}$ (10)

where $I_{1}$ is the single qubit identity matrix and $\alpha$ is a
coefficient determining its degree of polarization or mixedness. This state is
a locally thermal state with respect to the system Hamiltonian

$\displaystyle H_{C}=-\omega\sigma^{z}\,:\,\beta=\omega^{-1}\tanh(\alpha)$
(11)

and being a single qubit state we can associate an effective inverse
temperature $\beta$ to it where $H_{C}$ is a typical qubit Hamiltonian with
$\omega$ a constant related to the Larmor precession frequency on the Bloch
Sphere. The final state of the logical qubit in the trace estimation protocol
before measurement is

$\displaystyle\rho^{\prime}_{C}(\alpha)=\frac{I_{1}+\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}\sigma_{1}^{z}}{2}$
(12)

for the trace estimation of unitaries with only real contributions. This
constraint on the computational input ensure that the state is always locally
thermal with respect to $H_{C}$.

$\rho_{C}$$\rho^{\prime}_{C}$$\beta$$\beta$$H_{C}$$H_{C}^{{}^{\prime}}$isothermal$\rho_{C}^{\prime}$$\beta^{{}^{\prime}}$$H_{C}$$\mathcal{E}(\rho)$$\mathcal{E}_{R}(\rho)$
Figure 2: The setup for our operational analysis as discussed in [45, 47].

The Tasaki-Crooks Fluctuation theorem for the logical qubit in this algorithm
can now be investigated where we consider the reversed channel
$\mathcal{E}_{R}(\cdot)$, derived below, which models the reverse process the
system can fluctuate to. Using a two-point-projective-measurement scheme (TPM)
[48] on the setup presented in Figure 2, work distributions for the forward
and reverse processes are obtained allowing us to derive the Fluctuation
Theorem in this context. In Figure 2 we also depict an isothermal process as
is typical of setups for this derivation [45, 47], carried out by an external
agent with local control over the logical qubit who changes the Hamiltonian’s
precession frequency to

$\displaystyle\omega^{\prime}=\omega\left(\operatorname{arctanh}(\alpha)\right)^{-1}\operatorname{arctanh}\left(\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}\right).$
(13)

For a more extensive discussion on the association of an effective temperature
to these states refer to Appendix B.

It will prove useful to convert this channel to its Choi representation (see
Appendix C for a detailed derivation) [41, 49, 50]

$\displaystyle\Upsilon_{\mathcal{E}}=\left(\mathcal{E}(\rho)\otimes
I_{1}\right)|\varphi\rangle\\!\langle\varphi|$ (14)

where $\ket{\varphi}=\sum^{2}_{i=1}\ket{ii}$ i.e. the two qubit un-normalised
Bell state. The impact of a channel on a quantum state making use of its Choi
representation is given by
$\mathcal{E}\left(\rho_{S}\right)=\text{tr}_{B}\left\\{\left({I_{S}}\otimes\rho_{S}^{\text{T}}\right)\Upsilon_{\mathcal{E}}\right\\}$
so it is easy to verify that

$\displaystyle\mathcal{E}\left(I_{1}\right)=\text{tr}_{2}\left\\{\left(I_{1}\otimes{I_{1}}^{\text{T}}\right)\Upsilon_{\mathcal{E}}\right\\}=I_{1}$
(15)

where $\text{tr}_{2}\\{\cdot\\}$ refers to tracing over the second d.o.f,
meaning that the identity is a fixed point of this map and that this Choi
state represents unital dynamics, as expected from the result of Section 3. We
now define the time-reversed channel for this process as

$\displaystyle\mathcal{E}_{R}(\rho)=\text{tr}_{n}\left\\{V^{\dagger}\left(\rho\otimes\frac{I_{n}}{2}\right)V\right\\}$
(16)

which has a Choi state that is the complex conjugate of Eq. (14) and as such
is also unital giving

$\displaystyle\frac{\mathcal{P}_{F}(\Delta E_{C})}{\mathcal{P}_{R}(\Delta
E_{C})}=e^{\beta\left(\Delta E_{C}-\Delta F\right)}$ (17)

where $\mathcal{P}(\Delta E_{C})$ is an energy distribution obtained through
the TPM across the forward or backwards channel, as indicated by the
subscript, and $\Delta F$ is the free energy change in the isothermal process,
following the presentation of [45].

As the overall dynamics is unitary, we can analyse the energetic exchanges
between the logical qubit and the register. As shown in Appendix D, since the
register remains in a maximally mixed state throughout the process, there is
no heat exchange from the logical qubit to the environment and therefore

$\Delta E_{c}=\langle W_{C}\rangle.$ (18)

We find that the work done, $\langle W_{C}\rangle$, on the logical qubit is
solely due to the heat from the register in the trace estimation algorithm
giving

$\displaystyle\frac{\mathcal{P}_{F}(W_{C})}{\mathcal{P}_{R}(W_{C})}=e^{\beta\left(W_{C}-\Delta
F\right)}=e^{\Sigma_{C}}$ (19)

where $\Sigma_{C}$ is the entropy production of the logical qubit discussed in
Appendix D. By using the generalised Landauer equality [51, 52] in this
setting, cfr. Sec. D, we obtain the Tasaki-Crooks Fluctuation theorem for the
logical qubit

$\displaystyle\frac{\mathcal{P}_{F}(W_{C})}{\mathcal{P}_{R}(W_{C})}=e^{\Delta
S_{C}}.$ (20)

##  5 The thermodynamics of DQC1 trace estimation

Once more focusing on computational inputs with only real trace contributions,
the thermodynamics of the trace estimation algorithm is straightforward. In
trace estimation, the logical qubit being used as a control qubit, is out of
equilibrium with the maximally mixed ancillary register (which corresponds to
an infinite temperature state). The more polarised the logical qubit, as
quantified by $\alpha$, the more out of equilibrium it is with the ancillary
register. The algorithm allows for these two quantum systems to interact and
begin to equilibriate, with different trace estimations leading to different
degrees of equilibration. To analyse the thermodynamics at play we can use the
generalised Landauer equality or non-equilibrium 2nd Law [51, 52]

$\displaystyle\langle\Sigma\rangle=\beta\langle Q\rangle+\Delta
S=\mathcal{I}(S^{\prime}:R^{\prime})+D(\rho^{\prime}_{R}||\rho_{R})$ (21)

as presented in Appendix D where the relationship

$\displaystyle\Delta E_{C}$ $\displaystyle=-\langle Q_{A}\rangle=\langle
W_{C}\rangle$ (22) is derived and the subscripts $C$ and $A$ denote the
logical qubit and ancillary register respectively, giving
$\displaystyle\langle W_{C}\rangle$
$\displaystyle=\text{tr}\left\\{H_{C}(\rho^{\prime}_{C}-\rho_{C})\right\\}$
$\displaystyle=\alpha\omega\left(1-\frac{\Re\\{\text{tr}U\\}}{2^{n}}\right).$
(23)

We see that the heat exchanged with the logical qubit is directly dependent on
what trace is being estimated where the farther the trace being estimated is
from that of the identity, the more the energy changes. If the trace being
estimated is that of the identity, the Hamiltonian of the ancillary register
will not be changed by an external agent in the implementation of this
unitary. As a result, no heat is exchanged with the control qubit and so its
energy will not change. On the other hand, the trace estimation of some
generic unitary necessitates an external agent changing the Hamiltonian of the
ancillary qubits, doing work on these qubits. This change in Hamiltonian can
lead to interactions between the logical and ancillary qubits in trace
estimation, and so a change in its energy.

The pre-factor $\alpha$ acts as a temperature gradient, indicating that the
greater the temperature difference at the outset, the greater the possible
energy change through equilibration. The degree of equilibration is then being
constrained by the computational input as the system-bath interaction given by
the controlled-unitary clearly impacts the thermodynamics experienced by the
register.

Turning our attention to the von Neumann entropy
$S(\rho)=-\text{tr}\\{\rho\ln\rho\\}$, the entropy of the initial and final
state before measurement are

$\displaystyle S(\rho_{C})$
$\displaystyle=H_{2}\left(\frac{1-\alpha}{2}\right)$ $\displaystyle
S(\rho^{\prime}_{C})$
$\displaystyle=H_{2}\left(\frac{1-\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}\right)$

which are both binary entropy functions $H_{2}(x)=-(1-x)\ln(1-x)-x\ln(x)$. The
change in entropy, $\Delta S_{C}$, is shown in Figure 3. On examination, we
see that the entropy is in general increasing satisfying the 2nd law of
thermodynamics and that there is dependence on both the normalised trace being
estimated and the polarisation of the logical qubit. As discussed in Section
4, the dynamics of the control qubit respects the Tasaki-Crooks relation,
where the entropy production has contributions only from the change in entropy
of this qubit. From Figure 3 we see that the change in entropy has dependence
on the trace being estimated but that it is also significantly impacted by the
temperature of the logical qubit through $\alpha$. As such, the hotter the
logical qubit, the less entropy production and the more likely thermal
fluctuations become. We can corroborate this insight by looking at the work
statistics of the logical qubit, examining its nonequilibrium thermodynamics.

Figure 3: Change in entropy of the first qubit in the ($\Delta
S_{C}=S(\rho^{\prime}_{C})-S(\rho_{C}))$, for different normalised trace
contributions $\left(\frac{\Re\\{\text{tr}U\\}}{2^{n}}\right)$ and values of
polarisation $\alpha$.

(a) $\alpha=1$

(b) $\alpha=0.5$

(c) $\alpha=0$

Figure 4: Work distributions for the logical qubit in DQC1 trace estimation
with different values of $\alpha$. In these plots, $\theta$ is the parameter
from Eq. (28) giving different two qubit unitaries whose trace is estimated
and $\omega=1$. As $\alpha$ becomes smaller and so the effective temperature
of the logical qubit becomes larger, fluctuations become more apparent in the
work distribution.

A two-point-measurement scheme [48, 53] may be implemented to investigate the
work distribution associated to the logical qubit. Here we measure in the
energy eigenbasis at the start and end of the protocol allowing us to
establish a probability distribution associated to the work. Whilst
delineations between heat and work are often difficult to make, in our setting
this is not an issue due to Eq (18) meaning that all energy changes detected
in the logical qubit are due to work done on it, equal to the heat it receives
from the maximally mixed register.

The work probability distribution is given by

$\displaystyle\mathcal{P}(W)=\sum_{n,m}p_{n}p^{\prime}_{m|n}\delta\left(W-(E^{\prime}_{m}-E_{n})\right)$
(24)

where $p_{n}=\bra{n}\rho_{S}\ket{n}$ are the probabilities associated to the
initial state,
$p^{\prime}_{n|m}=\left|\bra{n}\rho^{\prime}_{S}\ket{m}\right|^{2}$ are the
conditional probabilities associated to the final state and $E_{m}$ and
$E_{n}$ are the energy eigenvalues associated to the initial Hamiltonian
$H=\sum_{n}E_{n}|E_{n}\rangle\\!\langle E_{n}|$ and the final Hamiltonian
$H^{\prime}=\sum_{m}E^{\prime}_{m}|E^{\prime}_{m}\rangle\\!\langle
E^{\prime}_{m}|$. In this setting, we have the same initial and final
Hamiltonian $H_{C}=-\omega\sigma_{z}$ for the logical qubit at the start and
end of the algorithm, before measurement. The probabilities achieved by the
scheme are readily evident from the initial state $\rho_{C}(\alpha)$ whose
populations give the initial probabilities and the final state
$\rho^{\prime}_{C}(\alpha)$ whose populations are conditional giving a
distribution with the first moment of the work distribution being

$\displaystyle\langle
W_{C}\rangle=\int^{\infty}_{-\infty}W_{C}\mathcal{P}(W_{C})\,dW_{C}=\Delta
E_{C}$ (25)

which is equal to the change in energy of the logical qubit, recovering Eq.
(23). Examining the second moment we find

$\displaystyle\text{Var}[W_{C}]$
$\displaystyle=\int^{\infty}_{-\infty}W^{2}_{C}\mathcal{P}(W_{C})\,dW_{C}$
(26)
$\displaystyle=2\omega^{2}\left(\frac{\Re\\{\text{tr}U\\}}{2^{n}}\alpha^{2}-1\right)$
(27)

showing that fluctuations in the work distribution are mainly dependent on
$\alpha$, since $\omega$ is a scaling term, corroborating what we expected
from the fluctuation theorem satisfied by the logical qubit, Eq. (20).

Different trace estimations will have different work distributions, with some
displaying relevant nonequilibrium behaviour on examining their higher moments
if the distribution is asymmetric. To make this more tangible, we consider the
trace estimation of a family of $2\times 2$ matrices given by the
parameterised iSWAP

$\displaystyle\text{iSWAP}(\theta)=\begin{pmatrix}1&0&0&0\\\
0&\cos(\theta/2)&i\sin(\theta/2)&0\\\ 0&i\sin(\theta/2)&\cos(\theta/2)&0\\\
0&0&0&1\\\ \end{pmatrix}$ (28)

where the trace is real therefore corresponding to an effectively thermal
transformation suitable to the analysis we have made, and it is a function of
$\theta$ given by $\frac{\Re\\{\text{tr}U\\}}{4}=\cos^{2}(\frac{\theta}{4})$.
In Figure 4 we plot the work distribution for different values of $\theta$
giving us an insight into how the nonequilibrium work varies for the trace
estimation of this family of unitaries. We also plot these work distributions
for different values of $\alpha$, showing how fluctuations in the work
distribution become more prevalent for higher temperatures of the logical
qubit, i.e. lower values of $\alpha$.

##  6 Discussion & Conclusions

In this work we have treated the DQC1 complexity class as an open quantum
system whose dynamics we captured through the formalism of quantum channels.
We have shown that these channels are unital and therefore respect the Tasaki-
Crooks theorem. We have used this framework to investigate how the
computational task of trace estimation can be related to the thermodynamics
experienced by the logical qubit in the DQC1 register, in both average and
non-equilibrium settings. Restricting our analysis to unitaries with real
trace, we have investigated a purely thermal process associated to changes in
populations in thermal states.

In this setting we have shown that the average work done is directly related
to the real part of the trace being estimated. More broadly, calculating the
non-equilibrium work distribution shows that the trace estimation of different
unitaries makes use of different trajectories in the evolution of the logical
qubit throughout the computation. We have further shown that the
thermodynamics of a system carrying out a computation impacts the result of
the computation itself and that that backwards processes become more likely
for less entropy being produced. This last result highlights the effect of
fluctuations of the system at the end of a computation, before measurement,
which effectively yield a different state than one would expect to measure.

We believe that it would be of interest to investigate the thermodynamics of
other complexity classes of quantum algorithms; the idea of treating the
dynamics required to carry out an algorithm as a channel can be
straightforwardly extended to other protocols. We note that frameworks have
also been developed to investigate the quantum resources a quantum channel is
capable of generating over a Hilbert space [54]. Applied together with the
approach we describe in this work, this could open up a new route for
exploring the thermodynamics of creating correlations in quantum algorithms,
as has been explored in [55] using catalytic thermal operations.

## Acknowledgements

The authors acknowledge fruitful discussions with and comments from Tony J. G.
Apollaro, Chris Cade, Bariş Çakmak, Sebastian Deffner, Nicolai Friis, Marcus
Huber and Philip Taranto. In particular JX thanks Gabriel Landi who provided
great insight into how to think about entropy production, Elisa Bäumer whose
encouragement led to this work being pursued and Mirko Consiglio for careful
checking of derivations at airports. JX is supported by the Endeavour
Scholarships Scheme a project which is part-financed by the European Social
Fund and an IPAS+ Grant under the project MITIQ - Quantum Information and
Thermodynamics in Ireland and Malta. SC is supported by the Science Foundation
Ireland Starting Investigator Research Grant “SpeedDemon” No. 18/SIRG/5508. JG
is funded by a Science Foundation Ireland-Royal Society University Research
Fellowship, the European Research Council Starting Grant ODYSSEY (Grant
Agreement No. 758403).

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## Appendices

##  Appendix A Examining Unitality

Consider a quantum operation described by a Unitary operation $U$ on a system
$\rho_{S}$ of dimension $d_{S}$ and an environment $\rho_{B}$ of dimension
$d_{B}$ given as

$\displaystyle\mathcal{E}(\rho_{S})$
$\displaystyle=\text{tr}_{B}\left\\{U\rho_{S}\otimes\rho_{B}U^{\dagger}\right\\}$
(29)
$\displaystyle=\sum^{d_{B}}_{i,j}\bra{i}U\left(\rho_{S}\otimes\sum^{d_{B}}_{j=1}b_{j}|j\rangle\\!\langle
j|\right)U^{\dagger}\ket{i}$ (30)
$\displaystyle=\sum^{d_{B}}_{i,j}b_{j}\bra{i}U\ket{j}\rho_{S}\bra{j}U^{\dagger}\ket{i}$
(31)

where $\ket{i},\ket{j}$ are eigenkets of the environment and $b_{j}$ are the
eigenvalues of the initial state of the environment. The Kraus representation
of this channel will then be expressed through Kraus operators of form

$\displaystyle K_{i,j}$ $\displaystyle=\sqrt{b_{j}}\bra{i}U\ket{j}$ (32)

allowing us to examine the Unitality of this channel through the expression

$\displaystyle\sum^{d_{B}}_{i,j}K_{i,j}K^{\dagger}_{i,j}=\sum^{d_{B}}_{i,j}b_{j}\bra{i}U|j\rangle\\!\langle
j|U^{\dagger}\ket{i}$ (33)

which we can examine by cases.

###  A.1 Separable Unitaries

A separable unitary can be split over the dimensions of the system and
environment as $U=U_{S}\otimes U_{B}$ so in (33) this gives

$\displaystyle\sum^{d_{B}}_{i,j}K_{i,j}K^{\dagger}_{i,j}$
$\displaystyle=\sum^{d_{B}}_{i,j}b_{j}\bra{i}U_{S}\otimes
U_{B}|j\rangle\\!\langle j|U_{S}^{\dagger}U_{B}^{\dagger}\ket{i}$ (34)
$\displaystyle=\sum^{d_{B}}_{i,j}b_{j}U_{S}\bra{i}U_{B}|j\rangle\\!\langle
j|U_{B}^{\dagger}\ket{i}U_{S}^{\dagger}$ (35)
$\displaystyle=\sum^{d_{B}}_{i,j}b_{j}U_{S}u_{i,j}u^{*}_{j,i}U_{S}^{\dagger}$
(36) $\displaystyle=I_{S}$ (37)

where we have used the property that the eigenvalues of density matrices sum
to 1 and that the rows and columns of unitary matrices form an orthonormal
basis. Seperable unitaries in a dilation give a unital channel for any
environment used in the dilation.

###  A.2 Controlled Unitaries

If the system of interest is a single qubit controlling a unitary operation
being carried over the environment in the dilation, we can show that the
resultant channel on this single qubit is unital independently of the initial
state of the environment. This is sensible as such a channel on a quantum
system being used to control a unitary operation on the environment being
traced out is the identity channel, making it trivially unital. We present the
below derivation as the the presentation is still insightful. A controlled
unitary can be expressed as

$\displaystyle|0\rangle\\!\langle 0|\otimes\sum^{d_{B}}_{k}|k\rangle\\!\langle
k|+|1\rangle\\!\langle 1|\otimes\sum^{d_{B}}_{l,m}u_{l,m}|l\rangle\\!\langle
m|$ (38)

and substituting in (33) we have

$\displaystyle\sum^{d_{B}}_{i,j}K_{i,j}K^{\dagger}_{i,j}$ (39)
$\displaystyle=\sum^{d_{B}}_{\begin{subarray}{c}i,j\\\ k,l,m,\\\
n,p,q\end{subarray}}b_{j}\bra{i}\left(|0\rangle\\!\langle
0|\otimes|k\rangle\\!\langle k|+|1\rangle\\!\langle 1|\otimes
u_{l,m}|l\rangle\\!\langle m|\right)|j\rangle\\!\langle
j|\left(|0\rangle\\!\langle 0|\otimes|n\rangle\\!\langle
n|+|1\rangle\\!\langle 1|\otimes u^{*}_{p,q}|p\rangle\\!\langle
q|\right)\ket{i}$ (40) $\displaystyle=\sum^{d_{B}}_{\begin{subarray}{c}i,j\\\
k,l,m,\\\ n,p,q\end{subarray}}b_{j}|0\rangle\\!\langle
0|\delta_{i,k}\delta_{k,j}\delta_{j,n}\delta_{n,i}+|1\rangle\\!\langle
1|\delta_{i,l}\delta_{m,j}\delta_{j,p}\delta_{q,i}u_{l,m}u^{*}_{p,q}$ (41)

summing over indices repeated indices gives

$\displaystyle\sum^{d_{B}}_{\begin{subarray}{c}l,m,\\\
p,q\end{subarray}}|0\rangle\\!\langle 0|+|1\rangle\\!\langle
1|\delta_{l,q}\delta_{m,p}u_{l,m}u^{*}_{p,q}=I_{1}$ (42)

finally, contracting $q$ with $l$ and $p$ with $m$ and using the
orthonormality of rows and columns of unitary matrices we have unitality.

###  A.3 Arbitrary Unitaries

We may express the unitaries in the eigenbasis of the system and the bath in
(33) giving

$\displaystyle=\sum^{d_{B}}_{i,j}\sum^{d_{B}d_{S}}_{k,l,m,n}b_{j}\bra{i}u_{k,l}|k\rangle\\!\langle
l||j\rangle\\!\langle j|u^{*}_{m,n}|m\rangle\\!\langle n|\ket{i}$ (43) and
splitting the system and bath contributions for the system-bath eigenkets we
have
$\displaystyle=\sum^{d_{B}}_{i,j}\sum^{d_{B}}_{\begin{subarray}{c}k_{B},l_{B},\\\
m_{B},n_{B}\end{subarray}}\sum^{d_{S}}_{\begin{subarray}{c}k_{S},l_{S},\\\
m_{S},n_{S}\end{subarray}}b_{j}u_{k,l}u^{*}_{m,n}\braket{i}{k_{B}}|k_{S}\rangle\\!\langle
l_{S}|\braket{l_{B}}{j}\braket{j}{m_{B}}|m_{S}\rangle\\!\langle
n_{s}|\braket{n_{B}}{i}$ (44) where $\alpha=\alpha_{S}+\alpha_{B}$ for
$k,l,m,n$
$\displaystyle=\sum^{d_{B}}_{i,j}\sum^{d_{B}}_{\begin{subarray}{c}k_{B},l_{B},\\\
m_{B},n_{B}\end{subarray}}\sum^{d_{S}}_{\begin{subarray}{c}k_{S},l_{S},\\\
m_{S},n_{S}\end{subarray}}b_{j}u_{k,l}u^{*}_{m,n}\delta_{i,k_{B}}\delta_{l_{B},j}\delta_{j,m_{B}}\delta_{n_{B},i}\delta_{l_{S},m_{S}}|k_{S}\rangle\\!\langle
n_{S}|$ (45) $\displaystyle=\sum^{d_{B}}_{\begin{subarray}{c}k_{B},l_{B},\\\
m_{B},n_{B}\end{subarray}}\sum^{d_{S}}_{\begin{subarray}{c}k_{S},l_{S},\\\
m_{S},n_{S}\end{subarray}}b_{l_{B}}u_{k,l}u^{*}_{m,n}\delta_{l,m}\delta_{k_{B},n_{B}}|k_{S}\rangle\\!\langle
n_{S}|$ (46) $\displaystyle=\sum^{d_{B}}_{\begin{subarray}{c}k_{B},l_{B},\\\
n_{B}\end{subarray}}\sum^{d_{S}}_{\begin{subarray}{c}k_{S},l_{S},\\\
n_{S}\end{subarray}}b_{l_{B}}u_{k,l}u^{*}_{l,n}\delta_{k_{B},n_{B}}|k_{S}\rangle\\!\langle
n_{S}|$ (47)

at which point we are unable to continue simplifying without knowing the form
of the unitary. In particular, what system-bath interactions it instantiates.
From this equation, we see that for a given unitary one would be able to find
an environment which results in a unital channel but a general result obtained
in this line of analysis seems unlikely and out of scope for this work.

##  Appendix B Effective temperature for the control qubit in the isothermal
case

A qubit whose energy eigenbasis is also its computational basis may be
described by the Hamiltonian

$\displaystyle H=-\omega\sigma^{z}$ (48)

and a thermal state relative to this Hamiltonian at inverse temperature
$\beta$ is expressed as

$\displaystyle\rho_{\beta}=\frac{e^{-\beta H}}{\text{tr}\\{e^{-\beta
H}\\}}=\begin{pmatrix}\frac{1+\tanh(\beta\omega)}{2}&0\\\
0&\frac{1-\tanh(\beta\omega)}{2}\end{pmatrix}.$ (49)

In our isothermal analysis of the trace estimation algorithm, the logical
qubit is taken to have an initial Hamiltonian $H_{s}=-\omega\sigma^{z}$ which
then changes to $H^{\prime}_{s}=-\omega^{\prime}\sigma^{z}$ as a result of
work done to implement the controlled unitary gate. We stipulate for the
isothermal case that the temperature of the controlled qubit remain fixed
giving the thermal states

$\displaystyle\rho_{\beta}=\frac{e^{-\beta H}}{\text{tr}\\{e^{-\beta
H_{s}}\\}}=\begin{pmatrix}\frac{1+\tanh(\beta\omega)}{2}&0\\\
0&\frac{1-\tanh(\beta\omega)}{2}\end{pmatrix}$
$\displaystyle\rho^{\prime}_{\beta}=\frac{e^{-\beta
H^{\prime}_{s}}}{\text{tr}\\{e^{-\beta
H^{\prime}_{s}}\\}}=\begin{pmatrix}\frac{1+\tanh(\beta\omega^{\prime})}{2}&0\\\
0&\frac{1-\tanh(\beta\omega^{\prime})}{2}\end{pmatrix}.$ (50)

The initial and final reduced states of the logical qubit before measurement
are diagonal in the energy eigenbasis

$\displaystyle\rho=\frac{I+\alpha\sigma^{z}}{2}=\begin{pmatrix}\frac{1+\alpha}{2}&0\\\
0&\frac{1-\alpha}{2}\end{pmatrix}$
$\displaystyle\rho^{\prime}=\frac{I+\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}\sigma^{z}}{2}=\begin{pmatrix}\frac{1+\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}&0\\\
0&\frac{1-\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}\end{pmatrix}$ (51)

and so take the form of the above thermal states leading to the relations

$\displaystyle\beta\omega=\operatorname{arctanh}(\alpha)$
$\displaystyle\beta\omega^{\prime}=\operatorname{arctanh}\left(\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}\right)$
(52)

where we have related the polarisation of the control qubit to the inverse
temperature and a constant related to the energy gap of the qubit’s basis
states. In this way, we can fix $\beta$ and $\alpha$ at the outset of the
process allowing us to formulate $\omega^{\prime}$

$\displaystyle\omega^{\prime}=\omega\left(\operatorname{arctanh}(\alpha)\right)^{-1}\operatorname{arctanh}\left(\alpha\frac{\Re\\{\text{tr}U\\}}{2^{n}}\right).$
(53)

such that an external agent with control over this parameter could carry out
work on the qubit taking it from the initial state $\rho$ to a final state
$\rho^{\prime}$ isothermally.

##  Appendix C Deriving the Choi Representation for the DQC1 Trace Estimation
Algorithm

Consider the circuit for the DQC1 trace estimation protocol where the
evolution of the state of the first qubit as a result of this protocol, before
measurement, can be modelled by the quantum operation

$\displaystyle\mathcal{E}(\rho)=\text{tr}_{n}\left\\{V\left(\rho\otimes\frac{I_{n}}{2}\right)V^{\dagger}\right\\}.$
(54)

where $V=(H\otimes I_{n})\left(|0\rangle\\!\langle 0|\otimes
I_{n}+|1\rangle\\!\langle 1|\otimes U\right)(H\otimes I_{n})$. By the Choi-
Kraus Theorem we can associate a Kraus representation to this map

$\displaystyle\mathcal{E}(\rho_{S})=\sum_{i}\Lambda_{i}\rho_{S}\Lambda^{\dagger}_{i}\,:\,\sum_{i}=\Lambda_{i}^{\dagger}\Lambda_{i}=I_{1}$
(55)

where $\Lambda_{i}$ are the associated Kraus operators. These Kraus operators
may be derived from the unitary $V$ in the Stinespring dilation as

$\displaystyle\Lambda_{i,j}=\bra{i_{B}}V\ket{\widetilde{j_{B}}}=\sqrt{b_{j}}\bra{i_{B}}V\ket{j_{B}}$
(56)

where $\ket{i_{B}}$ is an eigenket of the environment, $\ket{\tilde{j}}$ is a
eigenket for the inital state of the environment and $\ket{b_{i}}$ is an
eigenvalue of the initial state of the environment. The Kraus operators are
not unique and dependent on the choice of basis for the environment. For the
case of 2+1 DQC1 with the general two qubit unitary

$\displaystyle U=\sum^{4}_{i,j=1}u_{ij}|i\rangle\\!\langle
j|=\left(\begin{array}[]{cccc}u_{11}&u_{12}&u_{13}&u_{14}\\\
u_{21}&u_{22}&u_{23}&u_{24}\\\ u_{31}&u_{32}&u_{33}&u_{34}\\\
u_{41}&u_{42}&u_{43}&u_{44}\\\ \end{array}\right)\,:\,u_{ij}\in\mathbb{C}$
(61)

whose trace we are estimating, this gives the 16 Kraus operators using the
computational basis.

$K_{00}=\left(\begin{array}[]{cc}\frac{1}{4}\left(u_{11}+1\right)&\frac{1}{4}\left(1-u_{11}\right)\\\ \frac{1}{4}\left(1-u_{11}\right)&\frac{1}{4}\left(u_{11}+1\right)\\\ \end{array}\right)$ | $K_{01}=\left(\begin{array}[]{cc}\frac{u_{13}}{4}&-\frac{u_{13}}{4}\\\ -\frac{u_{13}}{4}&\frac{u_{13}}{4}\\\ \end{array}\right)$ | $K_{02}=\left(\begin{array}[]{cc}\frac{u_{12}}{4}&-\frac{u_{12}}{4}\\\ -\frac{u_{12}}{4}&\frac{u_{12}}{4}\\\ \end{array}\right)$
---|---|---
$K_{03}=\left(\begin{array}[]{cc}\frac{u_{14}}{4}&-\frac{u_{14}}{4}\\\ -\frac{u_{14}}{4}&\frac{u_{14}}{4}\\\ \end{array}\right)$ | $K_{10}=\left(\begin{array}[]{cc}\frac{u_{31}}{4}&-\frac{u_{31}}{4}\\\ -\frac{u_{31}}{4}&\frac{u_{31}}{4}\\\ \end{array}\right)$ | $K_{11}=\left(\begin{array}[]{cc}\frac{1}{4}\left(u_{33}+1\right)&\frac{1}{4}\left(1-u_{33}\right)\\\ \frac{1}{4}\left(1-u_{33}\right)&\frac{1}{4}\left(u_{33}+1\right)\\\ \end{array}\right)$
$K_{12}=\left(\begin{array}[]{cc}\frac{u_{32}}{4}&-\frac{u_{32}}{4}\\\ -\frac{u_{32}}{4}&\frac{u_{32}}{4}\\\ \end{array}\right)$ | $K_{13}=\left(\begin{array}[]{cc}\frac{u_{34}}{4}&-\frac{u_{34}}{4}\\\ -\frac{u_{34}}{4}&\frac{u_{34}}{4}\\\ \end{array}\right)$ | $K_{20}=\left(\begin{array}[]{cc}\frac{u_{21}}{4}&-\frac{u_{21}}{4}\\\ -\frac{u_{21}}{4}&\frac{u_{21}}{4}\\\ \end{array}\right)$
$K_{21}=\left(\begin{array}[]{cc}\frac{u_{23}}{4}&-\frac{u_{23}}{4}\\\ -\frac{u_{23}}{4}&\frac{u_{23}}{4}\\\ \end{array}\right)$ | $K_{22}=\left(\begin{array}[]{cc}\frac{1}{4}\left(u_{22}+1\right)&\frac{1}{4}\left(1-u_{22}\right)\\\ \frac{1}{4}\left(1-u_{22}\right)&\frac{1}{4}\left(u_{22}+1\right)\\\ \end{array}\right)$ | $K_{23}=\left(\begin{array}[]{cc}\frac{u_{24}}{4}&-\frac{u_{24}}{4}\\\ -\frac{u_{24}}{4}&\frac{u_{24}}{4}\\\ \end{array}\right)$
$K_{30}=\left(\begin{array}[]{cc}\frac{u_{41}}{4}&-\frac{u_{41}}{4}\\\ -\frac{u_{41}}{4}&\frac{u_{41}}{4}\\\ \end{array}\right)$ | $K_{31}=\left(\begin{array}[]{cc}\frac{u_{43}}{4}&-\frac{u_{43}}{4}\\\ -\frac{u_{43}}{4}&\frac{u_{43}}{4}\\\ \end{array}\right)$ | $K_{32}=\left(\begin{array}[]{cc}\frac{u_{42}}{4}&-\frac{u_{42}}{4}\\\ -\frac{u_{42}}{4}&\frac{u_{42}}{4}\\\ \end{array}\right)$
| $K_{33}=\left(\begin{array}[]{cc}\frac{1}{4}\left(u_{44}+1\right)&\frac{1}{4}\left(1-u_{44}\right)\\\ \frac{1}{4}\left(1-u_{44}\right)&\frac{1}{4}\left(u_{44}+1\right)\\\ \end{array}\right)$ |

Using the Kraus representation for $\mathcal{E}(\rho)$ provided by these
operators we may construct the Choi state corresponding to this channel by the
Choi-Jamiołkowski isomorphism [41, 49, 50]

$\displaystyle\Upsilon_{\mathcal{E}}=\left(\mathcal{E}(\rho)\otimes
I_{1}\right)|\varphi\rangle\\!\langle\varphi|$ (62)

allowing us to obtain

$\displaystyle\Upsilon_{\mathcal{E}}=\frac{1}{16}\begin{pmatrix}4+\sum^{4}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}&-\sum^{4}_{i,j}|u_{ij}|^{2}+i\Im\\{\text{tr}U\\}&-\sum^{4}_{i,j}|u_{ij}|^{2}+i\Im\\{\text{tr}U\\}&4+\sum^{4}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}\\\
-\sum^{4}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}&4+\sum^{4}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&4+\sum^{4}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&-\sum^{4}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}\\\
-\sum^{4}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}&4+\sum^{4}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&4+\sum^{4}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&-\sum^{4}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}\\\
4+\sum^{4}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}&-\sum^{4}_{i,j}|u_{ij}|^{2}i\Im\\{\text{tr}U\\}&-\sum^{4}_{i,j}|u_{ij}|^{2}+i\Im\\{\text{tr}U\\}&4+\sum^{4}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}\end{pmatrix}$
(63)

which can clearly scale to $n+1$ DQC1 trace estimation as follows

$\displaystyle\Upsilon_{\mathcal{E}}=\frac{1}{2^{n+2}}\begin{pmatrix}2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}&-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}+i\Im\\{\text{tr}U\\}&-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}+i\Im\\{\text{tr}U\\}&2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}\\\
-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}&2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}\\\
-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}&2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-\Re\\{\text{tr}U\\}&-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}\\\
2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}&-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}-i\Im\\{\text{tr}U\\}&-\sum^{2^{n}}_{i,j}|u_{ij}|^{2}+i\Im\\{\text{tr}U\\}&2^{n}+\sum^{2^{n}}_{i,j}|u_{ij}|^{2}+\Re\\{\text{tr}U\\}\end{pmatrix}.$
(64)

Finally, this Choi representation is only dependent on the trace contributions
since for unitary matrices we have that the square of the Frobenius norm

$\displaystyle{||U||_{F}}^{2}=\sum^{2^{n}}_{i,j=1}|u_{ij}|^{2}=\text{tr}\\{U^{\dagger}U\\}=\text{tr}\\{I_{n}\\}=2^{n}$
(65)

giving the Choi state as in (6)

$\displaystyle\Upsilon_{\mathcal{E}}=\begin{pmatrix}\frac{1+\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}&i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}&i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}&\frac{1+\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}\\\
-i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}&\frac{1-\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}&\frac{1-\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}&-i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}\\\
-i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}&\frac{1-\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}&\frac{1-\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}&-i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}\\\
\frac{1+\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}&i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}&i\frac{\Im\\{\text{tr}U\\}}{2^{n+1}}&\frac{1+\frac{\Re\\{\text{tr}U\\}}{2^{n}}}{2}\end{pmatrix}$
(66)
$\displaystyle\Upsilon_{\mathcal{E}}=\frac{1}{2}\left[\begin{array}[]{c|c}I_{1}+\frac{\Re\\{\text{tr}U\\}}{2^{n}}\sigma_{z}+\frac{\Im\\{\text{tr}U\\}}{2^{n}}\sigma_{y}&\sigma_{x}+i\frac{\Re\\{\text{tr}U\\}}{2^{n}}\sigma_{y}+i\frac{\Im\\{\text{tr}U\\}}{2^{n}}\sigma_{z}\\\
\hline\cr\sigma_{x}-i\frac{\Re\\{\text{tr}U\\}}{2^{n}}\sigma_{y}-i\frac{\Im\\{\text{tr}U\\}}{2^{n}}\sigma_{z}&I_{1}-\frac{\Re\\{\text{tr}U\\}}{2^{n}}\sigma_{z}-\frac{\Im\\{\text{tr}U\\}}{2^{n}}\sigma_{y}\end{array}\right]$
(69)

On examination, one finds that this is a rank-2 Choi Matrix meaning that this
map can be described by two unwieldy Kraus Operators.

##  Appendix D Two Energetic 1st Laws for DQC1

The main feature of DQC1 algorithms is that they make use of a register

$\displaystyle\frac{I_{1}+\alpha\sigma^{z}_{1}}{2}\otimes\frac{I_{n}}{2^{n}}$
(70)

where $\alpha=1$ in the original clean DQC1 case [22] and $\alpha\in[0,1]$ in
the non-clean case [34, 39], and it is only the first qubit which is measured.
In this sense, the first qubit is the logical or special qubit and the qubits
in the maximally mixed state may be thought of as ancillas. The letter $C$ is
used here as in the trace estimation setting the logical qubit acts as a
control qubit. Let us label the first qubit
$\rho_{C}=\frac{I_{1}+\alpha\sigma^{z}_{1}}{2}$ and the mixed qubits
$\rho_{A}=\frac{I_{n}}{2^{n}}$.

The equation

$\displaystyle\langle\Sigma\rangle=\beta\langle Q\rangle+\Delta
S=\mathcal{I}(S^{\prime}:R^{\prime})+D(\rho^{\prime}_{R}||\rho_{R})$ (71)

expresses the average entropy production in terms of the correlations formed
across a system and a reservoir, quantified by the mutual information, and
relative entropy of the states of the reservoir at the start and end of a
nonequilibrium process where initially, the reservoir is in a thermal state
and the system and reservoir are separable. This expression was originally
explored in [51] to split the entropy production into a thermodynamically
reversible contribution due to heat exchange and thermodynamically
irreversible contribution due to the formation of correlations but was later
reformulated [52] as presented above which can be regarded as nonequilibrium
2nd Law. Clearly, the register of DQC1 algorithms is amenable to this
framework so let’s start our thermodynamic analysis here.

From the perspective of the logical qubit, we can consider it to be the
thermodynamic system of interest and the maximally mixed register as the
reservoir, which is thermal relative to some
$H_{A}=\sum_{i}\omega_{i}\sigma^{z}_{i}$ local Hamiltonian. Here, we have
$D(\rho^{\prime}_{A}||\rho_{A})=0$ since
$\rho^{\prime}_{A}=\rho_{A}=\frac{I_{n}}{2^{n}}$ and the mutual information

$\displaystyle\mathcal{I}\left(\rho^{\prime}_{C}\,:\,\rho^{\prime}_{A}\right)=S(\rho^{\prime}_{C})+S(\rho^{\prime}_{A})-S(\rho^{\prime}_{CA}).$
(72)

Since von Neumann entropy is invariant under the action of a unitary we find

$\displaystyle S(\rho^{\prime}_{CA})=S(V\rho_{C}\otimes\rho_{A}V^{\dagger})$
(73) $\displaystyle=S(\rho_{C}\otimes\rho_{A})=S(\rho_{C})+S(\rho_{A})$ (74)

where $V\in SU(N+1)$ is the global unitary representing a given DQC1 algorithm
and the additivity of the von Neumann entropy over separable states was used
giving

$\displaystyle\mathcal{I}(\rho^{\prime}_{C}\,:\,\rho^{\prime}_{A})=S(\rho^{\prime}_{C})-S(\rho_{C})=\Delta
S_{C}$ (75)

which implies that the entropy production of the control qubit is equal to its
change in entropy and it transfers no heat to the maximally mixed register.

$\displaystyle\langle\Sigma_{C}\rangle=\Delta S_{C}$
$\displaystyle\beta_{A}\langle Q_{C}\rangle=0$ (76)

This can also be seen from the fact that the ancillary component of the
register remains in the maximally mixed state throughout the computation
giving

$\displaystyle\beta_{A}\langle
Q_{C}\rangle=\text{tr}\left\\{(\rho^{\prime}_{A}-\rho_{A})H_{A}\right\\}=0$
(77)

and the 1st law for the logical qubit

$\displaystyle\Delta E_{c}=\langle W_{C}\rangle.$ (78)

Shifting perspectives, we can now focus on the maximally mixed register as our
thermodynamic system of interest and consider the logical qubit, whose initial
state is thermal relative to $H_{C}=\omega\sigma_{z}$, to be the reservoir. We
begin with the entropy where we now have $\Delta S_{A}=0$ since the state of
the ancillary register is unchanged and all entropy production will be
associated to heat exchanged. For the same reason, the 1st law in this context
is

$\displaystyle\langle W_{A}\rangle=-\langle Q_{A}\rangle.$ (79)

The mutual information remains the same as (6) due to its symmetry but the
relative entropy term is now

$\displaystyle
D(\rho^{\prime}_{C}||\rho_{C})=-S(\rho^{\prime}_{C})+\text{tr}\left\\{\rho^{\prime}_{C}\text{ln}\rho_{C}\right\\}$
(80)

giving the entropy production for the ancillas

$\displaystyle\langle\Sigma_{A}\rangle=\Delta
S_{C}-S(\rho^{\prime}_{C})+D(\rho^{\prime}_{C}||\rho_{C})$ (81)
$\displaystyle=\text{tr}\left\\{(\rho^{\prime}_{C}-\rho_{C})\text{ln}\rho_{C}\right\\}=\Delta
E_{C}$ (82) $\displaystyle\implies\langle\Sigma_{A}\rangle=\beta_{C}\langle
Q_{A}\rangle=\Delta E_{C}$ (83)

and so the entropy production of the ancillary register is equal to the heat
it exchanges with the logical qubit, which is equal to its energy change. In
this sense, the thermodynamics of this complexity class is straightforward.

As a result of the equality form of Landauer’s principle [51, 52] the
thermodynamics of any DQC1 algorithm may be summarised as heat being exchanged
from the ancillary register to the logical qubit which varies depending on how
much the entropy of the first qubit changes. This will be dependent on
correlations formed between partitions involving the logical qubit and parts
of the ancillary register with the remaining qubits [35, 30] and the
temperature difference between the qubits inherent of the DQC1 complexity
class. Whilst the choice of what is the system and what is the bath is
subjective, the thermodynamics experienced in terms of the heat exchanged, is
not. As is verified by looking at both these perspectives.

It is important to appreciate that if one has DQC1 algorithms represented by
unitaries that are not energy preserving, that is

$\displaystyle[V,H_{S}+H_{C}]\neq 0$ (84)

that the work terms in (9) and (10) may include contributions from an external
agent implementing this unitary and more housekeeping would be necessary.
Since the ancillary qubits are in a maximally mixed state no work done on the
ancillary qubits by an external agent can be captured by this scheme. The work
we see is work carried out by the ancillary qubits in exchanging this heat
with the control qubit – which is equal to the energy change in the control
qubit. This cannot be said for the logical qubit in the DQC1 complexity class
where the work carried out on this qubit can not only feature heat it acquires
from the ancillary qubits through the interaction mediated by $V$ but also in
changes to its local Hamiltonian which may occur during the implementation of
$V$. For this reason, we dub these energetic 1st laws as opposed to standard
thermodynamic 1st laws.