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Quantum Computation of Finite-Temperature Static and Dynamical Properties of Spin Systems Using Quantum Imaginary Time Evolution

Shi-Ning Sun, Mario Motta, Ruslan N. Tazhigulov, Adrian T. K. Tan, Garnet Kin-Lic Chan, Austin J. Minnich

arXiv:2009.03542v1quant-ph

TL;DR

Finite-temperature quantum simulation can demand resources beyond near-term devices, while evidence for practical approaches remains limited. This paper applies QITE and resource-reduction and error-mitigation methods to spin systems of up to four sites on five-qubit IBM Quantum devices. It reports accurate two-site observables and four-site results within stated temperature ranges, while lower-temperature accuracy remains challenging.

  • Problem

    Finite-temperature algorithms for quantum many-body systems can require more qubits and deeper circuits than near-term devices can support.

    Source: §I · p. 1

  • Method

    The authors use ansatz-independent QITE with symmetry-based Pauli-string reduction, circuit optimization, and error mitigation to compute finite-temperature spin observables on quantum hardware.

    Source: §I · p. 1

  • Results

    Two-site static and dynamical observables are accurately captured across a wide temperature range, while four-site static results agree reasonably at β ≤ 0.5 and dynamical correlations are accurate at high temperature β ∼ 0.2.

    Source: §V · p. 10

  • Takeaways & Limitations

    With reduced quantum-resource demands and error mitigation, QITE enables practical calculations of finite-temperature energies, correlations, and excitation spectral densities on near-term hardware.

    Source: §V · p. 10

    For four-site systems, accurate lower-temperature observables remain challenging with the current scheme, which recompiles QITE unitaries separately at each imaginary-time step.

    Source: §V · p. 10

Abstract

from arXiv · show

Developing scalable quantum algorithms to study finite-temperature physics of quantum many-body systems has attracted considerable interest due to recent advancements in quantum hardware. However, such algorithms in their present form require resources that exceed the capabilities of current quantum computers except for a limited range of system sizes and observables. Here, we report calculations of finite-temperature properties including energies, static and dynamical correlation functions, and excitation spectra of spin Hamiltonians with up to four sites on five-qubit IBM Quantum devices. These calculations are performed using the quantum imaginary time evolution (QITE) algorithm and made possible by several algorithmic improvements, including a method to exploit symmetries that reduces the quantum resources required by QITE, circuit optimization procedures to reduce circuit depth, and error mitigation techniques to improve the quality of raw hardware data. Our work demonstrates that the ansatz-independent QITE algorithm is capable of computing diverse finite-temperature observables on near-term quantum devices.

I. INTRODUCTION

Finite-temperature simulation remains difficult on near-term quantum computers because proposed approaches can require deep circuits and many qubits, or depend on ansatz quality or assumptions. The paper presents QITE as an ansatz-independent route and reviews how it approximates imaginary-time evolution with iterative unitaries.

I. INTRODUCTION

  • Finite-temperature algorithms based on bath coupling or quantum Metropolis sampling require prohibitively many qubits and deep circuits for near-term devices.Variational alternatives depend on ansatz quality, while the cited minimal effective Gibbs ansatz presumes the eigenstate thermalization hypothesis.Source: §I · p. 1
  • QITE is ansatz-independent and approximates imaginary-time evolution with unitary operators acting on qubit domains.Combined with METTS sampling, it is used as the QMETTS algorithm for finite-temperature quantities; resource reduction and optimization remained underexplored.Source: §I · p. 1
  • The paper calculates finite-temperature energies, correlation functions, and excitation spectra for one-dimensional spin systems with up to four sites on five-qubit IBM Quantum devices.It combines symmetry-based Pauli-string reduction, circuit optimization, and error mitigation to obtain hardware results.Source: §I · pp. 1–2

A. Quantum imaginary time evolution (QITE)

  • QITE groups noncommuting local Hamiltonian terms into Trotter terms, then approximates each imaginary-time propagator with a unitary built from Pauli strings.At each step, measured operators form a linear system for the unitary coefficients; the state is propagated with the resulting unitary before the procedure repeats.Source: A. Quantum imaginary time evolution… · p. 2

B. Finite-temperature dynamical correlation functions

The section extends finite-temperature observable averaging to dynamical correlation functions. It inserts QITE evolution into an ancilla-based correlation circuit, then averages over initial states or evaluates traces to obtain thermal observables.

B. Finite-temperature dynamical correlation functions

  • Dynamical correlation functions at finite temperature can be computed by averaging the correlation function over initial states, extending the averaging procedure used for static observables.The circuit inserts QITE into a dynamical-correlation circuit and evolves the initial state to imaginary time β/2.Source: B. Finite-temperature dynamical… · pp. 2–3
  • The ancilla is initialized in |0⟩ and the system in |Ψ⟩; ancilla X and Y measurements yield the real and imaginary parts of the correlation function on the QITE-evolved state.A thermal average over initial states gives ⟨U(t)V⟩β.Source: Figure 1 · p. 3, B. Finite-temperature dynamical… · p. 3
  • For METTS initial states, an unweighted average yields the finite-temperature dynamical correlation function; alternatively, full or stochastic trace evaluation yields finite-temperature dynamical or static observables.Stochastic trace evaluation samples a subspace, whereas full trace evaluation samples the entire Hilbert space.Source: B. Finite-temperature dynamical… · p. 3

A. Pauli string reduction by Z2 symmetries

When the Hamiltonian and initial state share Z2 symmetries, QITE can restrict its Pauli strings while preserving the unitary's action on the state. Four-site examples show substantial string reductions without changing the energy trajectories.

A. Pauli string reduction by Z2 symmetries

  • If the Hamiltonian and initial state lie in a shared stabilizer sector, QITE's unitary action is equivalent using the full set or its symmetry-preserving Pauli-string subset.The reduced set uses strings in the normalizer quotient N(S)/S; real Hamiltonians and initial states additionally permit restricting to strings with an odd number of Y operators.Source: A. Pauli string reduction by Z2… · p. 3
  • For the four-site TFIM, one Z2 symmetry reduces strings across three D = 2 domains from 16 to 6.The example uses J = h = 1 and initial state |0001⟩.Source: Figure 2 · p. 4, A. Pauli string reduction by Z2… · p. 4
  • For the four-site Heisenberg model, two Z2 symmetries reduce the single D = 4 domain's strings from 120 to 6.The example uses J = Δ = 1 and initial state (|0101⟩+|1010⟩)/2.Source: Figure 2 · p. 4, A. Pauli string reduction by Z2… · p. 4
  • Reduced Pauli-string sets preserve the energy trajectories in both four-site examples, matching unreduced QITE and exact imaginary-time evolution.In the TFIM the count falls from 16 to 6; in the Heisenberg model it falls from 120 to 6.Source: A. Pauli string reduction by Z2… · p. 4, Figure 2 · p. 4

B. Circuit optimization

The optimization methods address circuit depths that would otherwise exceed near-term hardware capabilities. Two-site circuits can be reduced to constant depth, while four-site circuits are recompiled to reduce depth.

B. Circuit optimization

  • Two-site QITE and real-time evolution circuits can be optimized to constant depth, independent of the number of time steps.For real-time evolution, the Hamiltonian propagator is decomposed into six single-qubit gates and two CNOT gates.Source: B. Circuit optimization · p. 4
  • Four-site QITE and real-time circuits are recompiled because Trotterized circuits would be too deep for accurate implementation on existing devices.Each QITE unitary or real-time propagator is fitted to a parameterized circuit using a tensor-network-based quantum simulation library.Source: B. Circuit optimization · pp. 4–5
  • The four-site recompiled circuit starts with four U3 gates and adds alternating even-odd and odd-even rounds of CNOT and single-qubit gates.The illustrated circuit has three gate rounds.Source: Figure 3 · p. 5, B. Circuit optimization · p. 5
  • The recompiled unitary is selected by gradient descent to maximize fidelity, using Ry gates for real QITE unitaries and U3 gates for real-time evolution.The target and recompiled unitaries act on a reduced density operator from the finite domain.Source: B. Circuit optimization · p. 5

C. Error mitigation

The error-mitigation methods correct measurement data during QITE and correct the final dynamical correlation function. Simulated finite-temperature energies show that combining readout error mitigation with post-selection is most effective for the four-site case.

C. Error mitigation

  • Readout error mitigation and post-selection are applied at each QITE imaginary-time step, while phase-and-scale correction is applied once to the final dynamical correlation function.The correction divides the raw ⟨Z0(t)Z0⟩β by its raw value at t = 0 to enforce the known value of 1 there.Source: C. Error mitigation · pp. 5–6
  • Post-selection discards outcomes with stabilizer parities inconsistent with the initial state, as gate errors and decoherence can produce wrong-parity outcomes.For system-qubit measurements, CNOT gates can transform the stabilizer generator and commuting Pauli string for simultaneous measurement.Source: C. Error mitigation · p. 5
  • For four-site TFIM energies, readout error mitigation combined with post-selection is most effective; its advantage is not apparent in the two-site case.Both methods shift noisy data toward exact results in simulations using measurement sampling and the ibmq rome noise model.Source: Figure 5 · p. 6, C. Error mitigation · p. 6
  • Readout mitigation applies the inverse calibration matrix to measurement counts using least-square fitting, with full calibration on all 2^N initial states for N qubits.The procedure uses the built-in Qiskit readout error mitigation routine.Source: C. Error mitigation · pp. 5–6

IV. RESULTS

On five-qubit IBM devices, the two-site TFIM calculations recover finite-temperature static and dynamical observables across a range of temperatures. Hardware excitation spectra reproduce key exact features, with finite-time windows accounting for peak-frequency shifts.

A. Two-site calculations

  • The calculations used full trace evaluation on five-qubit IBM devices, with 8000 repetitions per Pauli-string measurement circuit.The system qubits were arranged adjacently on the five-qubit chain, with the ancilla nearest system qubit 0.Source: §IV · p. 6, A. Two-site calculations · p. 6
  • 1%–4% mean absolute percentage error separates hardware and exact two-site energy and static-correlation results, averaged over inverse temperatures.The calculations covered J = ±1, ±3 and h = 1.Source: A. Two-site calculations · p. 7, Figure 6 · p. 7
  • Hardware real and imaginary correlation components agree well with exact results at β = 0.2 and 1.8, even without phase-and-scale correction.The correlation was evaluated from β = 0 to 2 and from t = 0 to 8π.Source: A. Two-site calculations · p. 7, Figure 7 · p. 7
  • Hardware spectra place peaks at 0, ±5.94, and ±7.18, versus exact-diagonalization frequencies 0, ±6.00, and ±7.21; finite time explains the shifts.The exact spectrum was Fourier-transformed at the same real-time points as the hardware data.Source: A. Two-site calculations · pp. 7–8
  • As β increases, the hardware emission-peak amplitudes reproduce the trend toward ground-state population dominance over the first excited state near β = 0.4.The two emission frequencies are 7.18 and 5.94.Source: Figure 7 · p. 7, A. Two-site calculations · p. 8

B. Four-site calculations

Four-site hardware calculations show that recompiled QITE circuits improve static-observable agreement over Trotterized circuits, while static results remain depth- and noise-sensitive. With phase-and-scale correction, the hardware also reproduces four-site dynamical spectra, and ten stochastic samples match full-trace energy results.

B. Four-site calculations

  • Trotterized D = 2 QITE requires 12 CNOT gates per imaginary-time step, while recompiled circuits use three gate rounds; D = 4 Trotterization exceeds 50 CNOTs per step.The real-time propagator for dynamical correlations was separately recompiled with five gate rounds.Source: B. Four-site calculations · p. 8
  • Recompiled QITE observables are closer to exact results than Trotterized D = 2 results, but recompiled curves fail to track exact static observables after β = 0.5.At β = 0.5, five imaginary-time steps correspond to 15 CNOT layers, nearly the devices’ circuit-depth limit.Source: Figure 8 · p. 8, B. Four-site calculations · p. 8
  • 10 stochastic samples reproduce full-trace four-site energy results across β, matching evaluation over all 16 initial states.The comparison used recompiled D = 2 QITE unitaries.Source: Figure 9 · p. 9, B. Four-site calculations · p. 9
  • D = 4 recompiled results are not uniformly closer to exact values than D = 2, plausibly because the larger 28-dimensional linear systems are more affected by hardware noise.The comparison favors D = 4 for ⟨X0X2⟩β and ⟨X0X3⟩β, but not for energy or ⟨X0X1⟩β.Source: Figure 8 · p. 8, B. Four-site calculations · p. 9
  • Raw dynamical data can deviate substantially before correction: at t = 0, the hardware correlation is 0.821 + 0.397i rather than the exact value 1.The ancilla experiences only five gate rounds before measurement, despite up to 11 combined QITE and real-time rounds.Source: B. Four-site calculations · pp. 9–10
  • With phase-and-scale correction, four-site hardware spectra reproduce frequencies 0, ±4.90, ±6.37, and 7.84, as well as peak amplitudes.The dynamical calculation was at β = 0.2 using recompiled D = 2 QITE unitaries.Source: Figure 10 · p. 9, B. Four-site calculations · p. 9

V. CONCLUSION AND OUTLOOK

The hardware calculations cover two- and four-site finite-temperature observables, with accuracy depending on system size, temperature, and circuit treatment. Stochastic trace evaluation matched full trace evaluation on four sites in the studied temperature regime, while larger systems and lower temperatures remain challenging.

V. CONCLUSION AND OUTLOOK

  • On two sites, quantum hardware accurately captured static and dynamical observables across a wide temperature range, with constant-depth QITE and real-time circuits.The constant-depth real-time evolution reproduced exact finite-temperature dynamical correlation functions without phase-and-scale correction, unlike previous studies.Source: §V · p. 10
  • At four sites, recompilation yielded reasonable agreement with exact static observables for β ≤0.5, while accurate dynamical correlations were reproduced at β ∼0.2 using phase-and-scale correction.Accurate lower-temperature observables remained challenging when QITE unitaries were recompiled separately at each imaginary-time step.Source: §V · p. 10
  • Larger systems and lower temperatures will require additional circuit-depth reduction, lower device error rates, or scalable methods.Suggested routes include recompilation with merged imaginary-time steps and more efficient error mitigation or improved hardware.Source: §V · p. 10
  • On four sites, stochastic trace evaluation accurately reproduced full trace evaluation results in the studied temperature regime and has zero autocorrelation time compared with QMETTS.The authors identify further comparison of QITE with different sampling schemes as an open topic.Source: §V · p. 10

APPENDIX A: PROOF OF PAULI STRING REDUCTION BY Z2 SYMMETRIES

The proof shows how Z2 symmetries reduce the Pauli strings needed in QITE while preserving the relevant action on symmetry-sector states. It also distinguishes this reduction from qubit encoding when the QITE domain is smaller than the full system.

APPENDIX A: PROOF OF PAULI STRING REDUCTION BY Z2 SYMMETRIES

  • When a stabilizer commutes with the local Hamiltonian and the state lies in its symmetry sector, the QITE action can use Pauli strings from N(S)/S instead of all strings in the Hamiltonian.The resulting QITE unitary commutes with the stabilizer elements and preserves the symmetry sector.Source: Appendix A · pp. 10–11
  • Pauli strings outside N(S) have zero relevant expectation values, making the QITE linear system block diagonal and allowing those components to be set to zero.The proof uses commutation with the stabilizer and the assumption that the state belongs to its symmetry sector.Source: Appendix A · pp. 10–11
  • Strings in the same coset of N(S)/S have equivalent action on the symmetry-sector state, permitting a reduced set of representatives in the QITE unitary.The passage derives the reduction from the QITE unitary's terms for strings in the same coset.Source: Appendix A · p. 11
  • The Pauli-string reduction matches qubit encoding when D equals N, but differs for D < N because Clifford encoding can change string weights and increase QITE cost.For periodic boundary conditions with symmetry Z0Z1Z2Z3, X0Y3 becomes X0X1Y2 under encoding, requiring a larger QITE domain; the work therefore uses Z2 symmetries to reduce strings rather than eliminate redundant qubits.Source: Appendix A · p. 11

APPENDIX B: ERROR BARS IN TRACE EVALUATION

The appendix estimates uncertainty in finite-temperature observables from measurement sampling and, for stochastic trace evaluation, initial-state sampling. It expresses observables through evolved-state probabilities and expectations, then uses first-order variance propagation under stated independence assumptions.

APPENDIX B: ERROR BARS IN TRACE EVALUATION

  • For full trace evaluation, the variance is estimated by first-order expansion in the measured probabilities and observable expectations, assuming they are independent.The stated error is the square root of the variance.Source: Appendix B · p. 11
  • A finite-temperature observable is expressed as a sum of contributions from evolved basis states, each weighted by its unnormalized probability and observable expectation.The probability is the squared norm after imaginary-time evolution, and the expectation is evaluated on the corresponding evolved state.Source: Appendix B · p. 11
  • On quantum computers, each state's probability is built from energy expectation values at each imaginary-time step.The observable expectation is evaluated on the QITE-evolved state at β/2.Source: Appendix B · p. 11
  • For stochastic trace evaluation, the observable is written as N/D and its variance treatment includes initial-state sampling as well as measurement sampling, assuming N and D are independent.The appendix defines the stochastic numerator from sampled probabilities before expanding the ratio.Source: Appendix B · pp. 11–12
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