Preparing thermal states on noiseless and noisy programmable quantum processors

TL;DR

This work tackles the challenge of preparing Gibbs (thermal) states on quantum hardware, proposing two provable algorithms that avoid quantum phase estimation and avoid the pitfalls of variational methods. The ergodic ETH-based approach uses ancilla qubits as an effective infinite bath to drive the system toward the Gibbs state for ergodic systems, while the universal method employs monitored random circuits to approximate imaginary-time evolution for general Hamiltonians, at the cost of exponential postselection in the worst case. Together with error mitigation strategies and resource estimates, the authors validate their methods via simulations and IBM hardware experiments on models such as the hardcore Bose-Hubbard system, highlighting potential applicability to quantum chemistry, materials science, and quantum machine learning. The work advances practical Gibbs-state preparation on near-term devices and clarifies the trade-offs between ergodic-specific guarantees and general-purpose, resource-constrained sampling.

Abstract

Nature is governed by precise physical laws, which can inspire the discovery of new computer-run simulation algorithms. Thermal states are the most ubiquitous for they are the equilibrium states of matter. Simulating thermal states of quantum matter has applications ranging from quantum machine learning to better understanding of high-temperature superconductivity and quantum chemistry. The computational complexity of this task is hopelessly hard for classical computers. The existing quantum algorithms come with caveats: most either require quantum phase estimation rendering them impractical for current noisy hardware, or are variational which face obstacles such as initialization, barren plateaus, and a general lack of provable guarantee. We provide two quantum algorithms with provable guarantees to prepare thermal states on (near-term) quantum computers that avoid these drawbacks. The first algorithm is inspired by the natural thermalization process where the ancilla qubits act as the infinite thermal bath. This algorithm can potentially run in polynomial time to sample thermal distributions of ergodic systems -- the vast class of physical systems that equilibrate in isolation with respect to local observables. The second algorithm works for any system and in general runs in exponential time. However, it requires significantly smaller quantum resources than previous such algorithms. In addition, we provide an error mitigation technique for both algorithms to fight back decoherence, which enables us to run our algorithms on the near-term quantum devices. To illustration, we simulate the thermal state of the hardcore Bose-Hubbard model on the latest generation of available quantum computers.

Figures (5)

• Figure 1: Ergodic algorithm and its performance (a) Illustration of the ergodic algorithm. The qubits are divided into two parts, a system of $n$ qubits and an ancilla of $n_a$ qubits. The ancilla is reset to a thermal state at the beginning of each cycle; during the cycle both parts are weakly coupled by the Hamiltonian in Eq. \ref{['eq:methdo2_ham']}. (b) The layout of a lattice half-filled with hardcore bosons obeying the Hamiltonian in Eq. \ref{['eq:hamiltonian']} with $J=1$ and $U=0.1$. The circles represent the sites that can be occupied by a single boson, the edges connect the nearest neighbors. (c) The inverse gap $\Delta^{-1}$ of the underlying classical Markov process in our algorithm (Theorem \ref{['eq:ergodic_theorem2']}) for the system in (b) with $\beta = 1$, $\Omega=1$, and $\gamma=0.1$. The inverse gap grows linearly with the system size of the truncated system (as shown by the dashed line), where the truncation is performed in numerical order (i.e., a system of $n$ sites includes the sites $1,\dots,n$).
• Figure 2: Simulations and experiments. (a) and (b). Numerical simulations of a seven-qubit device including $n=4$ one-dimensional system qubits and $n_a = 3$ ancilla qubits for the target Hamiltonian in Eq. \ref{['eq:hamiltonian']}, where $J = U = 1$ . Panel (a) shows the probabilities of sampling Hamiltonian eigenstates for ergodic algorithm (left) and universal algorithm (right). The curves describe the $\beta = 1$ output averaged over $10^3$ samples for: noiseless circuits (blue circles), noisy circuits (green diamonds), and noise-optimized circuits (red squares). Dashed lines show the exact solution. The number of cycles in the Ergodic algorithm is $d=20$, $\gamma = 0.1$, and noise is modeled by single-qubit depolarizing channels after each cycle with the probability $p = \Gamma t$, where $t$ is the cycle time and $\Gamma = 10^{-3}$ is the noise rate. In the universal algorithm, we take $d=5$ cycles and take depolarizing noise which affects each qubit with probability $p =10^{-2}$ after 2-qubit gates and $p'=2\cdot 10^{-2}$ after 3-qubit gates (see Supplementary Section \ref{['Supp1:F']} for justification). (b) The expected energy of the output as the function of temperature in the same setting. (c) and (d) Implementation on IBM 7-qubit ibm_casablanca device. Histograms show the experimental sampling probabilities compared with the theoretical predictions. (c) Implementation of the ergodic algorithm for $n=1$ system qubit and $n_a = 1$ ancilla qubit for Hamiltonian $H = Z_1$. The coupling is $\lambda = 0.1$ for the first cycle and it decreases linearly with number of cycles, inverse average time is $\gamma = 0.01$. Experiment uses $150$ random circuit configurations with $8192$ samples per circuit. (d) Sampling probabilities for the universal algorithm utilizing $n_a = 3$ ancilla qubits and $n=2$ system qubits with the Hamiltonian $H = X_1X_2-Z_1-Z_2$. In the experiment we took $100$ random circuit configurations with $8192$ samples per circuit.
• Figure S1: Modes of operation for the universal algorithm. In mode 1, the circuit is randomized each time it is run. In mode 2, we run the same circuit a fixed number of times and randomize only after success. In mode 3, the circuit is randomly selected at the beginning of the experiment and remains fixed for the duration of the experiment.
• Figure S2: Universal algorithm with native gates. The circuit for the universal algorithm contains $d$ identical cycles; here we show only one cycle. In each cycle, three sub-cycles implement the terms $XX$, $YY$, $ZZ$, and $Z$, according to the decomposition in Eqs. \ref{['eqs:two-qubit_gate_decomposition']} and \ref{['eqs:three-qubit_gate_decomposition']}. Gates labeled $H$ and $S$ denote Hadamard gates and standard $S$ gates, gates labeled $\theta_k$ (here we have omitted the cycle label) implement unitaries $u_{km} = \exp(\frac{1}{2} i\theta_{k} v_m X)$, $v_m = \sqrt{\beta \alpha_m/d}$, $\alpha_m$ is the coefficient if front of the Hamiltonian Pauli term, and the symbol "$0$" next to a box indicates a postselected measurement.
• Figure S3: Additional simulations. (a)-(b) Simulation results for universal algorithm utilizing $n_a = 2$ ancilla qubits for 3-qubit Hamiltonian in Eq. \ref{['eqs:1d_qubit_system']}, $t = -2$, $U=4$ and $h_i = -1$ (Heisenberg model). The inverse temperature is $\beta = 1$ and number of cycles is $d=5$. Panel (a) shows the overlap between expected output state in Eq. \ref{['eq:expect_avrg_rho']} and the eigenstates of the Hamiltonian for noiseless circuit (blue circles), noisy circuit (green diamonds), noisy optimized circuit (red squares), and exact solution (dashed line). We use a depolarizing noise model with single-qubit error probability $p_2=0.01$ for 2-qubit gates and $p_3=0.02$ for 3-qubit gates. An averaged circuit output is computed for $10^3$ successful sample runs. (b) Output state energy as a function of temperature for the same setting as the panel (a). (c)-(d) Performance for ergodic algorithm for the same setting as Panel (a)-(b), except using $d=20$ cycles, $\gamma = 0.1$, and single-qubit depolarizing error probability $p = \Gamma t$ after each cycle, where $t$ is the cycle time and $\Gamma = 10^{-3}$. The average is based on $10^3$ sample runs.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• Theorem S1: formal of Theorem \ref{['eq:ergodic_theorem2']}
• Lemma S1
• Lemma S2
• Lemma S3
• Lemma S4
• Theorem S2: formal of Theorem \ref{['thm:main_text_universal']}
• Lemma S5