# Characterization and thermometry of dissipatively stabilized steady states

G. S. Grattan  A. M. Liguori-Schremp<EMAIL_ADDRESS>D. Rodriguez Perez
E. Kapit  Physics Department, Colorado School of Mines
1523 Illinois Street
Golden, CO 80401  W. Jones  P. Graf  National Renewable Energy Laboratory
15013 Denver W Pkwy
Golden, CO 80401

###### Abstract

In this work we study the properties of dissipatively stabilized steady states
of noisy quantum algorithms, exploring the extent to which they can be well
approximated as thermal distributions, and proposing methods to extract the
effective temperature T. We study an algorithm called the Relaxational Quantum
Eigensolver (RQE), which is one of a family of algorithms that attempt to find
ground states and balance error in noisy quantum devices. In RQE, we weakly
couple a second register of auxiliary “shadow” qubits to the primary system in
Trotterized evolution, thus engineering an approximate zero-temperature bath
by periodically resetting the auxiliary qubits during the algorithm’s runtime.
Balancing the infinite temperature bath of random gate error, RQE returns
states with an average energy equal to a constant fraction of the ground
state. We probe the steady states of this algorithm for a range of base error
rates, using several methods for estimating both T and deviations from thermal
behavior. In particular, we both confirm that the steady states of these
systems are often well-approximated by thermal distributions, and show that
the same resources used for cooling can be adopted for thermometry, yielding a
fairly reliable measure of the temperature. These methods could be readily
implemented in near-term quantum hardware, and for stabilizing and probing
Hamiltonians where simulating approximate thermal states is hard for classical
computers.

††preprint: Quantum Science & Tech. 2024 - RQE

##  I Introduction

Many important applications of quantum computation involve preparing and/or
approximating the ground state of quantum Hamiltonians, for which many
algorithms have been proposed, ranging from variational methods such as the
Variational Quantum Eingensolver (VQE) [1, 2] to techniques based on adiabatic
processes [3, 4] to the implementation of effective imaginary time evolution
[5, 6]. All these methods present substantial drawbacks or limitations,
including: dependency of the performance on the quality of the variational
ansatz [7, 8] and classical optimization to minimize the energy, in the case
of variational methods; limitation of the convergence time that grows without
bound for system size when phase transitions are encountered in the
trajectories of the adiabatic processes; necessity for intermediate steps of
tomography and costly classical post-processing for the imaginary time
evolution technique. On present and near-term, noisy quantum computers, all of
these problems pale in comparison to the issue of noise [9, 10, 11]. As
current and near-term quantum devices are irreducibly noisy, controlling the
strength and nature of the coupling of a quantum system to the environment is
of paramount importance in the development of quantum technology. Though at a
first glance counter-intuitive, coupling to the environment can actually yield
substantial advantages when the system-bath couplings and bath structure are
tuned appropriately [12, 13].

One might expect the coupling to the environment to increase the system’s
entropy, and thus to reduce its “quantum efficiency." And indeed, the
reduction of coupling to uncontrolled degrees of freedom of the environment
has enabled much of the progress in quantum information processing,
particularly in solid state systems [14, 15, 16]. However, carefully
engineering the bath coupled to a system in specific ways can effectively
reduce the system’s entropy. To this end, there are different approaches,
including cases in which the bath is effectively very cold and thus acts as an
entropy and/or energy “dump" [17, 18], or the Maxwell’s demon that is digital
error correction codes [19, 20]. Indeed, dissipatively stabilized states, and
thermal states in particular, are a novel alternative to fault tolerant error
correction in near term devices, with important theoretical and experimental
developments in the last few years [21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41]. Further, there is evidence that
these states can be very difficult to reproduce classically [42, 41, 43].

Significant questions remain about the detailed structure of these states, and
to find practical uses for them it is important to better understand them. An
effective way to study them is within the frame of quantum thermodynamics, an
emerging field of physics that unites concepts of quantum information with
concepts from thermodynamics such as entropy, work, heat, and temperature [44,
45, 46]. Some practical applications include, for instance, temperature
estimation and improved metrology methods within the context of quantum
thermometry using “impurity probes" in many-body systems [47, 48, 49, 50, 51,
52, 53, 54, 55].

In this work, using extensive numerical simulations and theory, we explore
dissipatively stabilized states within the framework of quantum
thermodynamics, by applying our algorithm to two 1D many-body Hamiltonians,
namely: the ferromagnetic Ising model with transverse field; and the
Heisenberg model on a ring. The algorithm presented here is called the
Relaxational Quantum Eigensolver (RQE) to be consistent with its first
proposal in [56, 57], and is formulated with near-term devices in mind. We
first show that the steady state reached by our algorithm returns an average
energy equal to a constant fraction of the ground state energy (assuming
random states have energy zero), which steadily improves as the error rate
decreases. Then, we find that in many cases, dissipatively stabilized states
obtained from our algorithm are well approximated by thermal states, with a
temperature set by balancing error rates. In general, we do not know a priori
that the dissipatively-stabilized states are thermal; indeed, undoubtedly
there are cases in which they would not be. However, for the two 1D many-body
Hamiltonians considered in our work, it is a reasonable assumption, which we
confirm through detailed simulations in this work. But even when the steady
state of the system is well-approximated as thermal [58, 59, 60, 45, 61], the
question of how to measure that temperature is nontrivial. This work presents
three methods to characterize the temperature of the dissipatively-stabilized
state obtained from our algorithm, one of which efficiently infers it from the
populations of the qubits used as the cold bath. Moreover, this new
thermometry technique is particularly resource efficient, as it requires only
a small bath while also providing a method to estimate a system’s temperature
without the need of computationally-expensive methods based on exact
diagonalization, or experimentally expensive processes such as full state
tomography.

In the remainder of this paper, we first describe the algorithm used to study
the steady state of two many-body Hamiltonians of interest. Then, we present
the results obtained to characterize the dissipatively-stabilized states,
first showing the approximation of the ground-state energy and then presenting
our thermometry studies. In doing so we benchmark our new method, quantify its
accuracy and identify potential mechanisms that can cause it to report overly
high temperatures (relative to the real effective temperature of the many-body
state). Finally, we draw conclusions and offer an outlook for future work.

##  II RQE Algorithm Description

To set up our RQE algorithm, we first choose a quantum many-body Hamiltonian
$H_{P}$ that governs the primary qubits and that we wish to minimize. We then
couple the primary qubits weakly to a second register of $N_{S}=N_{P}/2$
ancillary, or “shadow", qubits [17, 62, 34, 39]; the shadow qubits act as an
engineered bath that removes excitations from the “primary" system, acting as
an approximate error correction mechanism. The shadow qubit Hamiltonian
$H_{S}$ is given by

$H_{S}=\sum_{j=1}^{N_{S}}\frac{\omega_{S_{j}}}{2}\sigma^{z}_{S_{j}}$ (1)

with $\omega_{S_{j}}$ being the energy of the $j$-th shadow qubit; the
coupling between primary and shadow qubits is described by the Hamiltonian

$H_{PS}(t)=\sum_{jk}\Omega_{jk}(t)O_{P_{j}}O_{S_{k}}$ (2)

with $\Omega_{jk}(t)$ being the primary-shadow interaction energy coupling
primary qubit $j$ to shadow qubit $k$, and $O_{P_{j}}$, $O_{S_{k}}$ Pauli
operators acting on the $j$-th primary qubit and $k$-th shadow qubit,
respectively. Unless otherwise specified, the Pauli operators for the primary-
shadow interactions in this work are taken as
$\sigma^{y}_{P_{j}}\sigma^{y}_{S_{k}}$. The total system Hamiltonian is
$H_{T}(t)=H_{P}+H_{S}+H_{PS}(t)$.

In order to implement time evolution under the (total) Hamiltonian on a gate-
model quantum computer, we must Trotterize it with a small timestep $dt$. If
$dt$ is sufficiently small, though, we can approximate evolution by the so-
called Trotter decomposition

$e^{iH(t)dt}\ket{\psi}\approx\big{(}e^{iH_{P}dt}e^{iH_{S}dt}e^{iH_{PS}dt}+O(dt^{2})\big{)}\ket{\psi}$
(3)

This Trotter decomposition allows us to implement a continuous time evolution
Hamiltonian with a digitized, gate-based model. As we are focused on near term
NISQ implementations, higher order operation approaches [63] that can be
implemented to mitigate Trotter error are counterproductive here due to gate
error, and instead we use this simpler layering structure.

###  II.1 RQE algorithm details

A schematic of the RQE algorithm is shown in Figure 1: we take the initial
state of the composite system as
$\ket{\psi_{in}}\equiv\ket{\psi_{0,P}}\otimes\ket{0_{S}\cdots 0_{S}}$ with the
shadow qubits prepared in the totally polarized state $\ket{0\cdots 0}$. Then,
at time $t=0$, the following sequence of gates is applied (we refer to this
sequence as a single algorithmic layer)[56]:

  1. 1.

Evolve the system under $\ket{\psi}\rightarrow e^{-2i\pi dtH_{P}}\ket{\psi}$,
appropriately Trotterized;

  2. 2.

Apply the $Z$ rotations which define the shadow qubit Hamiltonian,
$\ket{\psi}\rightarrow e^{-2i\pi dtH_{S}}\ket{\psi}$; this can be done in
parallel with step (1);

  3. 3.

Apply the primary-shadow qubit interaction term, $\ket{\psi}\rightarrow
e^{-2i\pi dtH_{PS}(t)}\ket{\psi}$;

  4. 4.

If $t$ is a multiple of $t_{PS}$, then reset all the shadow qubits to their
ground state $\ket{0\cdots 0}$; by periodically resetting the auxiliary qubits
during the algorithm’s runtime we can effectively engineer an approximate
zero-temperature bath which balances the effects of the infinite temperature
bath from random gate error;

  5. 5.

If $t=t_{f}$, then halt and enact the appropriate gate sequence to measure
$H_{P}$; otherwise, update $t\rightarrow t+dt$ and return to step (1).

To study the thermal properties of the system, a thermometry step (shown
schematically in Figure 2) is implemented by performing a measurement at the
end of the last reset cycle and getting the shadow qubit population for state
$\ket{1}$ instead of resetting the shadow qubits to state $\ket{0\cdots 0}$.
This thermometry step is performed only at the end, not during the algorithm,
because the shadow qubit energy is being varied in sweeps, which is more
efficient for cooling, but makes it impossible to determine how much energy
was extracted from the primary system when a $\ket{1}$ is measured. This step
is used in the third method to evaluate the primary system’s temperature by
fitting the average temperature $\langle T\rangle$ from the Fermi-Dirac
distribution, as explained below, and requires multiple time evolutions. This
thermometry step is key to the analysis of the dissipatively stabilized states
obtained from our algorithm and we compare the thermometry results with two
other methods used as benchmarks to characterize the temperature of the steady
state of our RQE algorithm. The details of the three methods to estimate the
temperature are explained in the next section.

Figure 1: Schematic of the RQE algorithm (top) with high-level details of the
algorithm implementation through Trotterized evolution (bottom): the entire
bottom figure is inside each $RQE_{n}$ cycle. $H_{P}$ is the problem
Hamiltonian acting on the primary qubit register, $H_{S}(t)$ acts on the
shadow qubit register, and $H_{PS}(t)$ is the Hamiltonian governing the
interaction between our primary and shadow qubit registers. We trotterize the
total Hamiltonaian $H(t)=H_{P}+H_{S}(t)+H_{PS}(t)$ and evolve for some period
of time before resetting the shadow qubits to the $|0\rangle$ state. We can
tune terms in $H_{S}(t)$ and $H_{PS}(t)$ to optimize the transfer of energy
and entropy from our primary qubits to our shadow qubits, which are
subsequently reset to reduce the energy of the combined system.  Figure 2:
Schematic of the thermometry step where we sample shadow-qubit energies
$\omega_{Sj}$ from a uniform distribution (top left), run RQE, and measure the
shadow qubits rather than resetting them (bottom). By repeating this we can
extract a thermal distribution relating the probability of measuring an
excited shadow qubit to its corresponding energy (top right).

Figure 3:  Ratio between the energy expectation value $\langle H_{P}\rangle$
for the steady state of the RQE algorithm and the energy of the ground state
$E_{GS}$ of the antiferromagnetic Heisenberg chain, as a function of ring size
$N_{P}$ for $N_{r}=40$ (left) and number of resets $N_{r}$ for $N_{P}=10$
(right), for different values of the two-qubit gate error rate $p_{2}$. For
the problems studied in this work, random states have energy zero.

Figure 4:  Thermometry of the dissipatively stabilized Heisenberg chain.
Values of energy expectation values $\langle H_{P}\rangle$ as a function of
temperature $T$ (left plot); fidelity measure $F_{KL}$ as a function of
temperature $T$ (right plot); for both methods, the number of primary qubits
is $N_{P}=10$ and we show results for two values of error rate
$p_{2}=0.0001,\,0.001$ and two circuit depths ($N_{r}=10,\,40$).

The variational parameters of the RQE algorithm are:

  * •

Algorithm time variable $t$ and global timestep $dt$; the latter is defined
relative to the problem Hamiltonian: if in each timestep the system evolves
via an appropriately Trotterized $\ket{\psi}\rightarrow e^{-i\alpha H_{P}}$,
then $\alpha\equiv 2\pi dt$. Note that $dt$ must be chosen carefully to
balance Trotter error and gate error, so in this work, we chose $dt$ to be
small enough to eliminate Trotter error while large enough to be able to
meaningfully evolve the system [64];

  * •

Runtime of the algorithm $t_{f}$ which defines the number of algorithmic
layers of gates $N_{L}\equiv t_{f}/dt$;

  * •

Time duration of a primary-shadow interaction pulse, $t_{PS}$, in which the
primary-shadow interaction is ramped up and down before the shadow qubits are
measured or reset; this defines the number of algorithm layers per pulse
$N_{L,PS}\equiv t_{PS}/dt$. For the cycles per reset in this work, after
trying a few different options, we found that for our given $dt$, 20 was a
reasonable value yielding a sufficiently long evolution allowing for the
system to thermalize without proliferating error dominating the system. In
comparison to other work [39], we use longer cycles, the coupling in our
algorithm is ramped up and down (instead of kept constant), and we focus on
slightly lower levels of simulated noise;

  * •

Number of times the shadow qubits are reset to their ground state, $N_{r}$;

  * •

Shadow-qubits energies $\omega_{S_{j}}$: in our normal RQE algorithm, they are
chosen similarly to [34], performing a downhill sweep for these energy values
from 6 to 0.75 for $3/4$ of the evolution time and then holding the values at
0.75 for the remainder of the time. However, in a Thermometry step as seen in
Figure 2, we randomly sample shadow-qubit energies from a uniform
distribution, $\omega_{Sj}\sim U(1,6)$, enabling us to extract the
distributions shown in Figures 2 & 5;

  * •

Primary-shadow interaction energies $\Omega_{jk}(t)$: for this interaction
energy coupling primary-qubit $j$ to shadow-qubit $k$ we use a function that
smoothly ramps up and down, specifically implementing a dome function
$\Omega_{PS}(t)\equiv 4t(1-t)$, where $t\equiv(k+1)/(N_{LPS}+1)$ with $k$
being the Trotter step index before reset; in our simulations we take this
dome function to have a phase sum (i.e. area under the curve) of
$\sum_{t=0}^{t_{PS}}\Omega_{PS}(t)dt=\pi/4$;

  * •

Error rate per operation $p_{j}$: generally, this includes single-qubit error
rate $p_{1}$, two-qubit gate error rate $p_{2}$, measurement error rate
$p_{M}$, and reset error rate $p_{R}$. Typically $p_{1}\approx p_{2}/10$ as a
phenomenological model 111More structured error models, such as the
loss/dephasing model in transmon qubits, can potentially be corrected more
efficiently.; $p_{2}$ represents the error per composite two-qubit unitary,
e.g. $e^{i\gamma\sigma^{z}_{j}\sigma^{z}_{k}}$. In particular, in our
algorithm we simulate depolarizing noise by randomly choosing one qubit for
each 2-qubit gate appended and applying one of the Pauli matrices with
probability $p_{2}/3$.

The parameter choices of our algorithm balance algorithmic efficiency and
efficacy. The choices of $dt$, $N_{L,PS}$, and $N_{r}$ are directly related to
circuit depth, the simulation runtime, and Trotter error. The initial values
of many parameters were intuitively guided and later refined through
variational methods to achieve the results presented in this work. One of the
main key factors was finding the balance between the Trotter timestep $dt$ and
the number of layers in an RQE cycle $N_{L,PS}$ to minimize Trotter error
while minimizing the expectation value of the prepared state. In our work we
found that $dt=0.0667$ and $N_{L,PS}=20$ balanced these criteria fairly well,
though obviously the best choice depends on the problem Hamiltonian $H_{P}$,
base error rate and other details. Additionally, our parameter choices for the
shadow qubits and reset rates dictate how well our engineered bath can cool
the system. Similarly, the initial choices of the shadow qubit energies
$\omega_{S_{j}}$, reset count $N_{r}$, and coupling strengths $\Omega_{jk}(t)$
were theoretically guided and later varied during experimentation to optimize
algorithmic efficacy.

##  III Ground state approximation

We first consider the degree to which RQE is able to approximate the ground
states of these systems. The first system we consider is a ring of $N_{P}$
primary qubits in a 1D ferromagnetic Ising model with transverse field,
described by the primary Hamiltonian $H_{P}$

$H_{P}=-J\sum_{j=1}^{N_{P}}\big{(}\sigma^{z}_{j}\sigma^{z}_{j+1}+\sigma^{z}_{1}\sigma^{z}_{N_{P}}\big{)}-\kappa\sum_{j=1}^{N_{P}}\sigma^{x}_{j}$
(4)

with ferromagnetic energy scale $J$ (set to 1 for simplicity) and transverse
field strength $\kappa<1$.
The second system we consider is a ring of $N_{P}$ primary qubits in an
antiferromagnetic Heisenberg model, described by the primary Hamiltonian

$\displaystyle
H_{P}=J\sum_{j=1}^{N_{P}}\big{(}\sigma^{x}_{j}\sigma^{x}_{j+1}+\sigma^{x}_{1}\sigma^{x}_{N_{P}}+\sigma^{y}_{j}\sigma^{y}_{j+1}+\sigma^{y}_{1}\sigma^{y}_{N_{P}}+\sigma^{z}_{j}\sigma^{z}_{j+1}+\sigma^{z}_{1}\sigma^{z}_{N_{P}}\big{)}.$
(5)

In Figure 3 the ratio between the energy expectation value $\langle
H_{P}\rangle$ for the steady state of the RQE algorithm and the energy of the
ground state $E_{GS}$ of the 1D antiferromagnetic Heisenberg ring is plotted
as a function of the number of primary qubits $N_{P}$ (left) and as a function
of the number of resets $N_{r}$ (right). In general, for a random state the
energy is zero. The energy expectation value of the steady state of the RQE
algorithm approximates the ground-state energy with results improving for
decreasing error rate $p_{2}$, as expected. These results align well with
other recent works [34, 39], and improve both with increasing number of
primary qubits $N_{P}$ and with higher-depth circuits (i.e. increasing number
of resets $N_{r}$).

To understand why RQE both returns a constant fraction approximation of the
ground state energy, and does not reach the exact ground state for zero error
(for the parameters chosen), let us consider the algorithm as an approximation
to continuous time evolution. Random gate error introduces an error rate
$\Gamma_{P}$ that heats the system toward infinite temperature; the primary-
shadow interaction increases this to
$\Gamma_{P}\left(1+\epsilon\right)+\Gamma_{R}^{\prime}$, where the factor of
$\epsilon$ comes from increasing circuit complexity per timestep and
$\Gamma_{R}^{\prime}$ is a small off-resonant heating rate from the shadow
qubits themselves, which vanishes as $\Omega_{PS}\to 0$ but is nonzero for the
parameters chosen here (more on this below). Balancing this rate, the shadow
qubits induce an average excitation removal or cooling rate $\Gamma_{R}$ which
depends on the algorithm parameters and not gate error. Hence, with
appropriate tuning, we can obtain a substantial cooling rate
$\Gamma_{R}\gg\Gamma_{P}$ which should allow our primary system to eventually
thermalize and approximate the ground state of the corresponding many-body
Hamiltonian $H_{P}$. Maximizing $\Gamma_{R}$ is thus key and can be achieved
by leveraging the resonant transitions by varying the shadow-qubit energies
$\omega_{Sj}$ during the algorithm’s runtime. We can thus expect the long-time
residual energy to scale roughly as $\Delta E\approx
N_{P}\frac{\Gamma_{P}(1+\epsilon)+\Gamma_{R}^{\prime}}{\Gamma_{R}}$ and
therefore the energy of the steady state of the RQE algorithm to approximate
the ground-state energy to a constant fraction. In general, for some systems
such as quantum spin glasses, the time to equilibrate may be prohibitively
long [66, 67]; it is empirically fairly short here but this is not a universal
property of the algorithm.

Figure 5:  Logarithmically scaled shadow-qubit energy distributions, used to
infer the shadow-qubit temperature $T_{S}$. Antiferromagnetic Heisenberg Chain
(left); Ferromagnetic transverse field Ising model (TFIM) with $\kappa=0.75$
(center) and $\kappa=0.90$ (right); all with $N_{P}=10$, $N_{r}=40$, and
$p_{2}=0.0001$.

Figure 6:  Temperatures extracted from the three different methods (explained
in Section IV) plotted versus system size: $T_{F_{KL}}$ from the fidelity
measure $F_{KL}$; $T(E)$ from the energy expectation values; $T_{S}$ from the
shadow qubit populations. Ferromagnetic Transverse Field Ising Model with
$\kappa=0.75$ (top left) and $\kappa=0.90$ (top right); Antiferromagnetic
Heisenberg ring (bottom left); Antiferromagnetic XXZ with primary-bath
operator $O_{P_{j}}O_{S_{k}}\equiv\sigma^{y}_{P_{j}}\sigma^{y}_{S_{k}}$ and
$J_{y}=0.5$ (bottom right). All four systems are plotted for high depth
circuits, i.e. $N_{r}$ = 40, with time step $dt=0.0667$ and number of
algorithm layers per pulse $N_{L,PS}=20$; the primary-shadow qubit coupling is
$2:1$ and the shadow-qubit energies are randomly sampled from a uniform
distribution, $\omega_{S_{j}}\sim U(1,6)$. The data is from 1000 random
trajectories per point.

Figure 7: Temperature extracted using the three different methods (i.e., $T$
from the energy expectation value $\langle H_{P}\rangle$; $T$ from the
distribution that minimizes the fidelity measure $F_{KL}$ in the $Z$-basis;
$T$ from the shadow qubit populations) plotted versus system size: Here we
slightly modify the thermometry method by randomly selecting the primary-bath
operator $O_{P_{j}}O_{S_{k}}$ from
$\\{\sigma^{x}_{P_{j}}\sigma^{x}_{S_{k}},\sigma^{y}_{P_{j}}\sigma^{y}_{S_{k}},\sigma^{z}_{P_{j}}\sigma^{z}_{S_{k}}\\}$
and averaging over multiple samples (left); Temperature overestimation ratio,
$2T_{S}/(T_{F_{KL}}+T_{E(T)})$, plotted versus system size comparing our
traditional primary-bath operator
$O_{P_{j}}O_{S_{k}}\equiv\sigma^{y}_{P_{j}}\sigma^{y}_{S_{k}}$ to the randomly
sampled operator (right). Both plots are for TFIM with $\kappa=0.90$ and
$N_{r}=40$. The three thermometry methods as well as the overestimation are
discussed in Section IV.

##  IV Thermometry

We now turn to the thermometry of these dissipatively stabilized states. In
order to better understand the thermal behavior of the steady state obtained
from the RQE algorithm, we use three methods to characterize the temperature
of the dissipatively stabilized state, namely:

  1. 1.

Inferring $T$ from the energy expectation value $\langle H_{P}\rangle$;

  2. 2.

Inferring $T$ from the distribution that minimizes K-L divergence between the
true thermal density matrix and measurements of bitstrings in the $Z$-basis;

  3. 3.

Inferring $T$ from the shadow qubit populations.

While the first two methods are both based on exact diagonalization of the
primary system Hamiltonian, the third method is applicable on real quantum
hardware and to cases where exact diagonalization is not possible. The first
two methods are used to calibrate our expectations and check the results
obtained from our RQE algorithm with the third method: in fact, as shown in
Figure 6, the temperatures extracted from the three different methods are
broadly consistent with each other.

For the first method, in particular, we are assuming that the steady state is
thermal: while in general this is not always the case, for the two 1D many-
body Hamiltonians considered in our work, it is a reasonable assumption, which
is also confirmed by the results obtained with the K-L divergence method
presented in the next section. Thus, we can use an operational definition of
temperature (as in [45]), consider the system consisting only of primary
qubits, and fully diagonalize it. From the exact diagonalization, we obtain
$2^{N_{P}}$ energy eigenvalues $\\{E_{j}\\}_{j=1,\dots,2^{N_{P}}}$. Then, for
each temperature value $T$ in a given range, we compute the partition function
$Z=\sum_{j=1}^{2^{N_{P}}}e^{-E_{j}/T}$ and the thermal expectation value for
the energy as a function of the temperature $T$:

$E(T)=\frac{\sum_{j=1}^{2^{N_{P}}}E_{j}e^{-E_{j}/T}}{Z}$ (6)

Hence we extrapolate at what specific temperature value $T^{*}$ the curve
$E(T)$ crosses the energy expectation value $\langle H_{P}\rangle$ obtained
from the RQE algorithm. Moreover, we can also use this result from the exact
diagonalization to compare it with the temperature inferred from the shadow-
qubit populations, as explained in the third method below.

For the second method, we start by measuring the distance between the final
state obtained from the RQE algorithm and the thermal distribution obtained
from the transverse-field Ising model with a fixed number of primary qubits on
a ring. We measure this distance with the Kullback-Leibler divergence (a.k.a.
K-L divergence or relative entropy), with which one can measure how close, or
similar, one statistical distribution $P$ is to a given, or reference,
statistical distribution $Q$ [68, 69]. In our specific case, the distributions
are discrete; the given, or reference, distribution $Q$ can be taken to be the
thermal state obtained from exact diagonalization of the primary qubits
Hamiltonian; thus, the $Z$-basis distribution obtained from the RQE algorithm
is the distribution $P$ for which we want to calculate the distance from the
reference distribution $Q$.
We use the standard definition of the K-L divergence:

$D_{KL}(P\;||\;Q)\equiv\sum_{j=1,\dots,2^{N_{P}}}=P(j)\log\bigg{(}\frac{P(j)}{Q(j)}\bigg{)}$
(7)

From this definition, as in [43], we want to find the temperature (Figure 4)
which maximizes the normalized fidelity measure

$F_{KL}\equiv
1-\frac{D_{KL}(\rho_{therm}||\rho_{RQE})}{D_{KL}(\rho_{IUR}||\rho_{RQE})}$ (8)

where $\rho_{therm}=e^{-H/T}/\text{Tr}(e^{-H/T})$ is the ideal thermal state,
$\rho_{RQE}$ is the dissipatively stabilized state obtained from our
algorithm, and $\rho_{IUR}$ is the random uniform distribution where all
strings have probability $2^{-N_{P}}$. The normalization of the fidelity
measure $F_{KL}$ is performed by the incoherent uniform random distribution
since this is the distribution we would get in a high depth circuit with no
error correction. We use this method as a second check of the system’s
temperature since it is closer to what can be done on hardware, though it is
expensive as it requires many samples to converge (and requires us to
calculate $\rho_{therm}$ for comparison). However, it is much more efficient
than full state tomography once the system size is large. As shown in Figure
4, the temperatures extracted from this method and from the average energy
show excellent quantitative agreement, confirming the thermal character of
these states.

Finally, with the third method we estimate the temperature of the steady state
obtained from the RQE algorithm by fitting the average temperature $\langle
T\rangle$ from the number of shadow qubits in the excited state $\ket{1}$
using the Fermi-Dirac distribution

$\langle
n\rangle=\frac{e^{-\omega_{S_{j}}/(k_{B}T)}}{1+e^{-\omega_{S_{j}}/(k_{B}T)}}$
(9)

Since the shadow qubits act as a bath draining energy from the system of
primary qubits by extracting excitations, the number of shadow qubits found in
the state $\ket{1}$ at the end of the algorithm can be used as a measure for
the temperature of the primary system (given fixed shadow qubit
$\omega_{S_{j}}$).

We apply each of the three methods both to the 1D ferromagnetic Ising model
with transverse field and to the antiferromagnetic Heisenberg ring. For both
models, we performed simulations with different circuits depths (i.e. with the
number of resets ranging from $N_{r}=10$ up to $N_{r}=40$), for error rates
$p_{2}=0,\,0.0001,\,0.001$, and for rings with a varying number of primary
qubits $N_{P}=[4,\dots,13]$. For the Ising model, we considered two different
values of transverse field strength $\kappa=0.75,\,0.9$.

In Figure 4 we show results from the first two (i.e. the benchmark) methods
for the Heisenberg model. As can be seen here, $F_{KL}$ as defined in (8)
reaches a maximum value proving that the state reached with the RQE algorithm
in fact approaches the thermal distribution. Moreover, the temperature for
which the normalized K-L divergence reaches its maximum value is consistent
with the temperature $T(E)$ obtained from the energy expectation values. It is
important to note that for the TFIM the temperature does not reach an exactly
zero value. One of the physical causes of this is the nature of the
excitations in the system. In fact, the elementary excitations in the
ferromagnetic phase are non-local and topological, since they are domain
walls, and as such they can only be created and destroyed in pairs, in the
bulk [34, 35, 60]. Thus, topological excitations cannot be removed by local
operations; therefore, two or more of such excitations need to coincide in
order to be removed. This makes it effectively harder to cool quantum systems
with non-local (topological) excitations and challenging to prepare
topological ground states on quantum simulators.

For the third method, we have performed simulations for different ring sizes,
with primary qubit range $N_{P}=[4,\dots,13]$, varying the number of resets
$N_{r}$ of the shadow qubits and error rates $p_{2}=0.0001,\,0.001$. In Figure
5 we show the logarithmically scaled shadow-qubit energy distributions, used
to infer the shadow-qubit temperature $T_{S}$ for three cases, all with
$N_{P}=10$, $N_{r}=40$, and $p_{2}=0.0001$: antiferromagnetic Heisenberg ring
(left); ferromagnetic transverse field Ising model (TFIM) with $\kappa=0.75$
(center) and $\kappa=0.90$ (right). These extrapolated temperature values are
fairly consistent with those obtained from the other two methods, as shown
also in Figure 6. In particular, for the 1D ferromagnetic Ising model with
transverse field we see a relatively constant overestimation of the
thermometry temperature, $T_{S}$, which remains consistent with fixed error
rates over varying circuit depths; whereas for the antiferromagnetic
Heisenberg ring with $N_{P}>6$ we get more accurate predictions of the
thermometry temperature, $T_{S}$, improving as system size increases.

An intuitive explanation for the temperature overestimation can be given based
on concepts from [12]. As the interaction strength isn’t strictly zero, the
probability of the primary-shadow interaction inducing “off resonant"
transitions (e.g. interactions where $dE_{P}+dE_{S}\neq 0$) is also non-zero.
The rate $\Gamma_{\pm}(\Delta)$ of those processes is suppressed only at low-
order polynomially in the energy mismatch $\Delta$ and it is symmetric about
$\Delta=0$ [12]:

$\Gamma_{\pm}(\Delta)=\frac{\Omega_{jk}^{2}\omega_{S_{j}}/2}{(\Omega_{jk}^{2}\pm\Delta)^{2}+\omega_{S_{j}}^{2}/16}$
(10)

In contrast, the expected shadow-qubit population $\langle n\rangle$ vs. $E$
(as a function of temperature) is an exponential distribution; therefore, if
we add some “uncertainty" from such resonance effects, then that blurring is
going to symmetrically increase the expected population above a given energy
just as much as it increases the population below it, which in turn is going
to artificially increase $T$ in fitting. Furthermore, gate error which excites
a shadow qubit will lead to a higher temperature measurement (though with
$p_{1}=p_{2}/10$ this will be rare). Thus, on balance we expect that this
method will tend to overestimate the temperature.

This argument does not explain however why the shadow-qubit estimated
temperatures display a much larger relative overestimation for the TFIM than
in our Heisenberg simulations (see Figure 6). While we expect the topological
character of elementary excitations plays a role here, it turns out the choice
of operator ($Y_{j}$ by default) used for the system-bath coupling also
contributes significantly to the overestimation. Specifically, the TFIM in our
notation is composed of $ZZ$ interactions with a transverse field along $X$,
whereas the Heisenberg chain is isotropic and its Hamiltonian does not have
any “preferred directions" in operator space. While given an infinite time the
primary and shadow qubits should perfectly thermalize (assuming, of course,
the primary system is thermal in the first place), our thermometry method only
collects excitations for a single cycle. It stands to reason that if
particular operator choices bias the transfer of excitations (as a function of
energy or any other parameter), then this may contribute significantly to
“error" in thermometry from short time measures such as ours.

In order to address the difference in the accuracy of the temperature
extrapolated from the thermometry method for the TFIM vs. Heisenberg model, we
performed simulations for the XXZ model keeping all parameters the same as for
the Heisenberg ring except for decreasing the exchange energy between nearest-
neighbors in the $y$ direction, i.e. $J_{x}=J_{z}=1$ but $J_{y}=0.5$. In this
case, as seen in the lower right panel of Figure 6, the accuracy of the
temperature extrapolated from the thermometry method is not as close as the
Heisenberg model but better than the TFIM. And in order to further confirm
these expectations, we performed simulations for the TFIM with $\kappa=0.9$
randomly sampling the primary-shadow interaction operators
$O_{P_{j}}O_{S_{k}}$ from
$\\{\sigma^{x}_{P_{j}}\sigma^{x}_{S_{k}},\sigma^{y}_{P_{j}}\sigma^{y}_{S_{k}},\sigma^{z}_{P_{j}}\sigma^{z}_{S_{k}}\\}$,
as shown in Figure 7: as can be clearly seen here, the estimation of the
temperature inferred from the RQE thermometry method improves when randomly
sampling the primary-shadow interaction operators $O_{P_{j}}O_{S_{k}}$ from
$\\{\sigma^{x}_{P_{j}}\sigma^{x}_{S_{k}},\sigma^{y}_{P_{j}}\sigma^{y}_{S_{k}},\sigma^{z}_{P_{j}}\sigma^{z}_{S_{k}}\\}$
vs. $O_{P_{j}}O_{S_{k}}\equiv\sigma^{y}_{P_{j}}\sigma^{y}_{S_{k}}$.

We present all these simulations to note and carefully illustrate how operator
choice is a potential source of error in thermometry through this method. We
note, of course, that the choice of primary-shadow interaction operator is
another variational parameter for our algorithm, and the choice which is best
for cooling the primary qubits (we chose $Y_{j}$ because it led to the lowest
energies in our early simulations used to set parameters) may not be the best
for accurately measuring the temperature using the dissipative elements.

##  V Conclusions & Outlook

In this work, we characterized the dissipatively stabilized states of our
relaxational quantum eigensolver (RQE) algorithm by estimating their
temperature using three different methods which yield broadly consistent
results both for the 1D ferromagnetic Ising model with transverse field and
for the antiferromagnetic Heisenberg ring. The first two methods, minimizing
the normalized K-L divergence between the distribution obtained by $Z$ basis
measurements and a thermal average, and estimating temperature from the mean
energy, require exact diagonalization of the system (or similar
computationally expensive methods) to create the ideal distributions to which
our dissipative algorithm results are compared. They are thus not suitable for
acheiving advantage in quantum simulation, but are a useful tool to
characterize these states at small scales and set expectations, and both
measures returned temperatures that were very close to each other (and likely
within sampling error) in all cases, showing that the steady states of these
two problems are indeed thermal to good approximation.

To provide a more efficient measure of the temperature, we introduced a
thermometry method based on the dissipative elements themselves (“shadow
qubits" in the language of our algorithm). We showed that while this method
tends to overestimate the temperature on general grounds, and care must be
taken in operator choices to reduce this effect, it nonetheless can provide
qualitatively and often quantitatively good thermometry results (using the
first two methods as a benchmark). This thermometry method is simple and
scalable, and could be implemented in real quantum hardware to measure the
temperature of dissipatively stabilized steady states at beyond classical
scales. For instance, it could be applied to non-stoquastic primary
Hamiltonians $H_{P}$ (where Quantum Monte Carlo has a sign problem), where
getting thermal states can be classically hard. Moreover, given that this new
thermometry technique uses only $O(N_{P})$ shadow qubits (i.e. a small
external bath), one of its key values is resource efficiency; it represents a
novel use of the quantum resources of the system by providing an approximate
error correction mechanism that can be used for other purposes too.

Moreover, we also calculated the ratio between the energy expectation value
$\langle H_{P}\rangle$ for the steady state of the RQE algorithm and the
energy of the ground state $E_{GS}$ for the models studied, showing that the
energy of the steady state of the RQE algorithm reaches a constant fraction of
the ground-state energy, with results improving for decreasing error rate
$p_{2}$, as expected, and in good agreement previous works [34, 39]. Our
results improve both with increasing number of primary qubits $N_{P}$ and with
higher-depth circuits (i.e. increasing number of resets $N_{r}$),
demonstrating that we approached the asymptotic steady states of these systems
in reasonably short order. These results also suggest that this method could
prove to be an efficient shortcut to simulating thermal distributions in fault
tolerant machines. All that said, it is important to emphasize that even in
the noise-free case (where there was no error in the primary qubits), the
temperature did not reach zero in our algorithm as formulated, due to stray
excitations created by off-resonant interactions between the primary and
shadow qubits, and the topological character of excitations in the TFIM
specifically.

In future work, it would be interesting to further explore the fine tuning of
algorithm parameters with both noise and system size, though the fact that our
results converged to fixed approximation ratios and average temperatures
suggest that scaling a single parameter set for each model and error rate
works well in practice. More illuminating would be the application of this
thermometry algorithm in real quantum hardware, and in systems with more
complex geometries. We only explored one-dimensional problems in this work to
ensure we could get decent scaling estimates from numerical simulations, but
our algorithm can be applied in any dimension. On real quantum hardware with
long ranged connectivity (either naturally or through error corrected
encodings), this method could even be applied to infinite range problems such
as quantum spin glasses. In the worst case, the thermalization timescales in
such systems are expected to be very long at large $N$, but with rich
attendant physics that could be probed in new ways through our methods.

## Acknowledgements

We would like to thank Erez Berg, Vadim Oganesyan, Eleanor Rieffel, David
Schuster, Norm Tubman and Paul Varosy for valuable discussions of the issues
in this work. This work was supported by the Superconducting Quantum Materials
and Systems Center (SQMS) under contract number DE-AC02-07CH11359. The SQMS
Center supported EK’s advisory role in this project, and ALS and GG’s
theoretical and computational research. This work was also supported by the
National Science Foundation through grant PHY-1653820, and the Army Research
Office through Grant No. W911NF-17-S0001.

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