Adiabatic preparation of thermal states and entropy-noise relation on noisy quantum computers

1. Problem Statement

The simulation of finite-temperature equilibrium states (thermal states) on quantum computers is a significant challenge. While ground state preparation is well-established via algorithms like Quantum Adiabatic Evolution (QAE) or Quantum Phase Estimation, preparing finite-temperature Gibbs states ($\rho \propto e^{-\beta H}$) is more difficult.

• Limitations of existing methods: Current approaches (imaginary time evolution, Lindbladian dynamics, variational algorithms) often require extensive resources, large numbers of measurements, or efficient low-energy state preparation, making them impractical for near-term hardware.
• Physical constraints: Preparing the exact global Gibbs state via unitary evolution is physically unrealistic for closed systems, as unitary evolution preserves global entropy. Furthermore, the spectrum of many-body systems contains exponentially close energy levels, making exact global thermalization extremely slow.
• The Goal: The authors propose a method to prepare states that are locally at thermal equilibrium (i.e., reduced density matrices of subsystems match the Gibbs state) rather than requiring the exact global Gibbs state.

2. Methodology

The core of the proposed protocol is a generalization of adiabatic ground state preparation to finite temperatures, leveraging the property of thermalization.

A. Theoretical Framework

1. Adiabatic Evolution: The system starts in a thermal Gibbs state of a simple, easily preparable initial Hamiltonian $H_0$ (e.g., a transverse field $H_0 = -\sum X_j$) at an inverse temperature $\beta_0$.
2. Interpolation: The system evolves adiabatically under a time-dependent Hamiltonian $H(t) = (1-t)H_0 + tH_f$ to a target Hamiltonian $H_f$.
3. Local Thermalization Assumption: The protocol assumes $H(t)$ satisfies the thermalization property (eigenstate thermalization hypothesis). Under slow adiabatic evolution ($T \to \infty$), the system remains locally in thermal equilibrium at all times.
4. Isentropic Evolution: A key theoretical result derived by the authors is that in the thermodynamic limit, adiabatic evolution preserves the entropy density of local reduced density matrices.
   • While global entropy is conserved by unitary evolution, local entropy usually changes. However, the authors prove that for adiabatic paths between gapped Hamiltonians, the change in local entropy density scales with the boundary area of the subsystem ($|\partial A|/N$), vanishing as $N \to \infty$.
5. Temperature Mapping: Because entropy density ($S$) is conserved, the final inverse temperature $\beta_f$ can be determined from the initial state parameters and the final energy density ($E_f$):
   $$\beta_f = \frac{\partial_{\beta_0} S_0}{\partial_{\beta_0} E_f}$$
   This allows the calculation of the final temperature without knowing the exact final state, provided the initial entropy and final energy are known.

B. Handling Noise (The Mirror Circuit Protocol)

Real quantum hardware introduces noise, which increases entropy and breaks the isentropic condition. The authors propose a benchmarking method to quantify this:

1. Mirror Circuit: Instead of running the full evolution to $H_f$, the circuit runs forward to $t=T/2$ and then backward (inverting the gates) to return to $t=0$.
2. Entropy Measurement: In a noiseless setting, the system returns exactly to the initial product state. In a noisy setting, the system returns to a mixed state.
3. Local Observable: Since the initial state $H_0$ is a product state in the $X$-basis, the entropy density of the final noisy state can be estimated simply by measuring the expectation value of the local $X$ observable ($\langle X \rangle$) on the qubits.
4. Noise Resilience: The authors show numerically that while noise increases the energy and entropy individually, the resulting Energy-Temperature ($E$ vs. $\beta$) curve remains robust. A noisy preparation at a target temperature is equivalent to a noiseless preparation at a slightly higher temperature.

C. Adiabaticity Check

To verify if the evolution is sufficiently adiabatic on hardware, the authors propose inserting a few Trotter steps of the final Hamiltonian $H_f$ in the middle of the mirror circuit. If the evolution is adiabatic, the state remains thermal, and the entropy estimate via $\langle X \rangle$ remains consistent regardless of the insertion. If not adiabatic, the entropy estimates diverge.

3. Key Contributions

1. Relaxed Objective: Shifted the goal from preparing exact global Gibbs states to locally thermal states, which is physically achievable via unitary evolution and sufficient for most condensed matter applications.
2. Entropy Conservation Proof: Provided a derivation showing that ideal adiabatic evolution preserves local entropy density in the thermodynamic limit, enabling temperature determination via energy and initial entropy.
3. Noise Benchmarking Protocol: Developed a "mirror circuit" technique to measure the entropy injected by hardware noise using simple local observables ($\langle X \rangle$), avoiding full state tomography.
4. Noise Resilience Discovery: Demonstrated that the $E(\beta)$ relationship is remarkably insensitive to depolarizing noise amplitude, meaning the protocol yields reliable temperature estimates even on noisy intermediate-scale quantum (NISQ) devices.
5. Hardware Implementation: Successfully executed the protocol on Quantinuum's H1-1 ion-trap device.

4. Results

• Numerical Simulations:
  • Verified that entropy density remains constant during adiabatic evolution in the thermodynamic limit (extrapolated from finite sizes).
  • Confirmed that the $E(\beta)$ curves for 1D Ising models collapse onto a single curve across different noise levels, proving noise resilience.
  • Validated the mirror circuit method for accurately estimating entropy growth due to noise.
• Hardware Experiment (Quantinuum H1-1):
  • System: 2D Ising model on a $5 \times 4$ lattice (20 qubits).
  • Circuit Depth: 640 two-qubit gates.
  • Measurements:
    • Measured entropy per site: $S = 0.166 \pm 0.0045$.
    • Measured energy density: $E = -2.334 \pm 0.0186$.
    • Derived final temperature: $T = 2.56 \pm 0.26$ (in units of the coupling constant).
  • Analysis: The measured temperature corresponds to roughly half the energy gap of the system. The discrepancy between the measured energy and the theoretical noiseless energy was attributed to leakage errors (qubits leaving the computational subspace), which affect energy measurements but do not necessarily increase the entropy of the measurement outcomes in the same way depolarizing noise does.

5. Significance

• Practical Thermal State Preparation: Offers a viable, resource-efficient pathway to prepare thermal states on NISQ hardware without requiring complex error correction or massive sampling overheads.
• Benchmarking Tool: The protocol provides a new metric for characterizing quantum hardware. By measuring the "effective temperature" and entropy injection, researchers can benchmark the noise resilience and adiabaticity of their specific device.
• Understanding Noise: The work highlights that while noise degrades state fidelity, the thermodynamic relationship between energy and temperature is robust, suggesting that thermodynamic properties can be reliably extracted from noisy quantum simulations.
• Future Directions: The authors identify leakage errors as a primary source of discrepancy in current hardware and suggest that future devices with leakage detection capabilities will further validate this approach. They also note that runtime scaling with system size and temperature remains an open question for optimization.

In summary, this paper establishes a theoretically grounded and experimentally verified method for preparing and characterizing finite-temperature states on noisy quantum computers, turning the limitations of current hardware into a feature for benchmarking thermal properties.