⚛️ quantum physics

Variational Thermal State Preparation on Digital Quantum Processors Assisted by Matrix Product States

This paper introduces a variational framework combining matrix product states and hardware-efficient ansätze to efficiently prepare quantum Gibbs states, demonstrating its scalability through numerical simulations on systems up to 6x6 sites and its practical viability on a 156-qubit IBM quantum processor with significant error mitigation.

Original authors: Rui-Hao Li, Semeon Valgushev, Khadijeh Najafi

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rui-Hao Li, Semeon Valgushev, Khadijeh Najafi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to bake the perfect loaf of bread, but instead of flour and water, you are working with quantum particles. In the world of quantum physics, these particles don't just sit still; they dance in complex patterns. When a system is "hot," the particles are chaotic and jittery. When it's "cold," they settle into a calm, organized rhythm.

Scientists call this organized, calm state a Gibbs state (or a thermal state). Creating these states on a quantum computer is like trying to bake that perfect loaf in a kitchen that is currently on fire (noisy, error-prone hardware). It's incredibly hard.

This paper introduces a clever new recipe to bake these quantum loaves without burning the kitchen down. Here is how they did it, explained simply:

1. The Problem: The "Entropy" Bottleneck

To bake the perfect thermal state, you need to minimize something called Free Energy. Think of Free Energy as a scorecard with two parts:

Energy: How much the particles are moving (like the heat in the oven).
Entropy: How messy or disordered the particles are (like the flour dusting everywhere).

The problem is that on a quantum computer, calculating the "messiness" (Entropy) is a nightmare. It usually requires taking a "snapshot" of the entire quantum system, which is like trying to photograph a hummingbird in a hurricane. It takes too much time and resources, especially on today's noisy computers.

2. The Solution: The "Digital Twin" (Matrix Product States)

The authors came up with a brilliant workaround. Instead of trying to measure the messiness directly on the quantum computer, they built a digital twin of the system on a regular classical computer.

The Analogy: Imagine you are trying to predict the weather. Instead of sending a million sensors into a storm to measure every gust of wind (which is hard and expensive), you build a highly accurate computer simulation of the storm. You run the simulation to see how the wind behaves, then use that data to guide your real-world actions.
The Tech: They used a mathematical tool called Matrix Product States (MPS). Think of MPS as a super-efficient compression algorithm. It can describe a complex quantum system using a tiny amount of memory, much like how a ZIP file compresses a large video.

By using this "digital twin," they could calculate the "messiness" (Entropy) instantly and accurately on a normal computer, without needing the quantum computer to do the heavy lifting.

3. The Recipe: Two Different Ovens (Ansatzes)

To create the state, they used a "quantum circuit" (a recipe of gates). They tested two different types of recipes:

Recipe A (TFDA): This is like a high-end, professional oven that starts with a perfectly mixed dough. It works great if you want a hot, chaotic loaf (high temperature), but it struggles to make a cold, dense loaf (low temperature) because it requires too many steps.
Recipe B (HEA - Hardware Efficient Ansatz): This is a simpler, sturdier oven designed for home kitchens. It starts with a plain, unmixed dough. Surprisingly, it turns out to be much better at making cold, dense loaves (low temperatures). Since most interesting physics happens when things are cold, this was the winner.

4. The Test Drive: From Simulation to Reality

The team didn't just stop at theory. They put their method to the test in two ways:

The Simulation: They simulated systems with up to 30 particles in a line (1D) and a 6x6 grid (2D). They checked if their "digital twin" could predict the temperature, pressure, and behavior of the system.
- Result: It worked beautifully! They could predict the behavior of these systems with high accuracy, even for sizes that are usually too big for classical computers to handle easily.
The Real Hardware: They took their best recipe and ran it on a real quantum computer at IBM (the IBM Heron processor with 156 qubits).
- The Challenge: Real quantum computers are noisy. It's like trying to bake bread while someone is shaking the table and blowing on the oven.
- The Fix: They used "Error Mitigation" techniques. Imagine you know your oven runs 5 degrees too hot, so you adjust the dial down. They used a technique called Zero-Noise Extrapolation (ZNE). They ran the experiment at different "noise levels" and mathematically extrapolated what the result would have been if the oven were perfect.
- Result: This cut their errors in half! They successfully prepared a thermal state for a 30-particle system, proving that their method works on real, imperfect hardware.

The Big Picture

Why does this matter?

Material Science: We can simulate how materials behave at different temperatures to discover new superconductors or batteries.
AI: This helps train "Quantum Boltzmann Machines," a type of AI that learns from data in a way similar to how our brains learn.
Optimization: It helps solve complex puzzles (like logistics or finance) by finding the best solution through "thermal sampling."

In a nutshell:
The authors built a hybrid system where a classical computer acts as a "smart assistant" to calculate the messy parts of the math, while the quantum computer acts as the "chef" that actually cooks the state. By using a smart, simple recipe (HEA) and a noise-canceling technique (ZNE), they managed to bake a perfect quantum loaf on a very noisy, real-world oven. This is a major step toward making quantum computers useful for understanding the real world.

1. Problem Statement

The preparation of quantum Gibbs (thermal) states at finite temperatures is a fundamental challenge in quantum computing, essential for simulating condensed matter systems, training quantum Boltzmann machines, and solving optimization problems.

Computational Hardness: Preparing ground states is QMA-hard; low-temperature Gibbs state preparation is believed to be equally demanding.
The Entropy Bottleneck: Variational Quantum Algorithms (VQAs) for thermal states typically minimize the Helmholtz free energy, $F(\rho) = E(\rho) - \beta^{-1}S(\rho)$ . While the energy term $E(\rho)$ is easily estimated via quantum measurements, calculating the von Neumann entropy $S(\rho)$ is resource-intensive, usually requiring full state tomography or stochastic reconstruction, which is infeasible on near-term noisy intermediate-scale quantum (NISQ) devices.
Scalability Issues: Existing variational approaches are limited to small systems (2–8 qubits) due to circuit depth constraints and the inability to efficiently evaluate entropy on hardware.

2. Methodology

The authors propose a hybrid classical-quantum framework that leverages Matrix Product States (MPS) to overcome the entropy bottleneck and enable scalable Gibbs state preparation.

A. MPS-Assisted Variational Algorithm

Instead of measuring entropy on the quantum device, the algorithm uses a classical MPS simulation to evaluate the cost function:

Purification: The mixed thermal state $\rho$ is represented as a pure state $|\psi\rangle$ on an enlarged Hilbert space (physical qubits + ancilla qubits) such that $\rho = \text{Tr}_{\text{anc}}(|\psi\rangle\langle\psi|)$ .
Circuit Parameterization: A parameterized quantum circuit (ansatz) prepares the pure state $|\psi(\theta)\rangle$ .
Classical Evaluation: The state $|\psi(\theta)\rangle$ $∣ ψ (θ)⟩$ is mapped to an MPS representation classically.
- Energy: Computed by contracting the MPS with the Hamiltonian (represented as a Matrix Product Operator).
- Entropy: Computed efficiently via the Schmidt decomposition of the MPS across the physical-ancilla bipartition. The eigenvalues of the reduced density matrix are the squared Schmidt coefficients ( $p_i = \lambda_i^2$ ), allowing $S(\rho) = -\sum p_i \ln p_i$ to be calculated without quantum measurements.
Optimization: A classical optimizer (COBYLA) iteratively updates parameters $\theta$ to minimize the free energy.
Execution: Once optimal parameters $\theta^*$ are found, the circuit is executed on a real quantum device to prepare the approximate Gibbs state and measure observables.

B. Ansatz Selection: HEA vs. TFDA

The paper compares two ansatzes:

Thermofield Double Ansatz (TFDA): Physically motivated, starting from a maximally entangled state. It performs well at high temperatures but requires deep circuits and many layers to disentangle systems at low temperatures, making it less suitable for NISQ devices.
Hardware-Efficient Ansatz (HEA): Composed of local rotations and entangling gates compatible with device connectivity.
- Finding: The HEA outperforms TFDA at low temperatures (large $\beta$ ). Since low-temperature thermal states have lower entanglement (following the area law), the shallow HEA can effectively approximate them. The authors select the HEA for large-scale simulations.

3. Key Contributions

Novel Framework: Integration of MPS for classical entropy evaluation within a variational Gibbs state preparation loop, bypassing the need for quantum tomography.
Scalability: Demonstrated the ability to simulate and prepare Gibbs states for 1D systems with up to 30 sites and 2D systems with up to $6 \times 6$ sites (using up to 44 qubits in simulation).
Hardware Demonstration: Successfully executed the preparation of a 30-site Transverse-Field Ising Model (TFIM) Gibbs state on a 156-qubit IBM Heron processor (ibm_kingston).
Error Mitigation: Applied Zero-Noise Extrapolation (ZNE) to hardware results, reducing relative errors in energy and susceptibility measurements by over 50% compared to unmitigated results.

4. Results

A. Noiseless Simulations (Classical)

Models Tested: 1D and 2D Transverse-Field Ising Model (TFIM) and 1D XXZ model.
Observables: Energy density, magnetic susceptibility, specific heat, and two-point correlation functions.
Performance Trends:
- Low Temperature ( $\beta > 2$ ): The HEA achieves high fidelity, with energy estimates closely matching exact analytical results (1D) or Quantum Monte Carlo (QMC) benchmarks (2D).
- Intermediate Temperature ( $\beta \approx 1$ ): Accuracy dips. This regime involves a complex mixture of many eigenstates with comparable weights, challenging limited-depth ansatzes.
- High Temperature: TFDA performs better, but HEA remains viable with sufficient ancilla qubits.
Operator Locality:
- Energy: Highly accurate across temperatures.
- Susceptibility: Less accurate than energy in 1D at intermediate temperatures due to its long-range nature (sum of all two-point correlations).
- Specific Heat: The least accurate observable. It depends on the variance of the Hamiltonian (a 4-local operator), making it highly sensitive to state preparation errors and requiring higher precision.

B. Quantum Hardware Experiments

Setup: 30-spin 1D TFIM on IBM Heron with $L=3$ layers and 4 ancilla qubits.
Error Mitigation:
- Techniques tested included Pauli twirling, Probabilistic Error Cancellation (PEC), and Dynamic Circuits.
- ZNE was the most effective. By folding two-qubit gates to amplify noise ( $\lambda = 1, 3, 5$ $λ = 1, 3, 5$ ) and extrapolating to the zero-noise limit, the authors achieved:
  - Energy: ~5.6% average relative error (down from ~11.4%).
  - Susceptibility: ~19.1% average relative error (down from ~38.9%).
Observation: Susceptibility estimates remained more error-prone than energy estimates, confirming the sensitivity of long-range operators to noise.

5. Significance and Outlook

Bridging the Gap: This work demonstrates a practical pathway to preparing thermal states on NISQ devices for system sizes previously inaccessible (30+ qubits), moving beyond the 2–8 qubit limit of prior works.
Efficiency: By offloading the difficult entropy calculation to classical MPS simulations, the method avoids the "measurement overhead" that plagues other variational thermal state methods.
Future Directions:
- Intermediate Temperatures: The method struggles at $\beta \sim 1/\Delta$ (where thermal energy matches excitation gaps). Future work may require problem-informed ansatzes (e.g., initializing with DMRG states) or deeper circuits with better error mitigation.
- Tensor Network Tomography: The authors suggest using MPS tomography to reconstruct states directly from hardware measurements, potentially bypassing the need for classical circuit simulation of deep circuits.
- Algorithmic Improvements: Addressing "barren plateaus" in deeper circuits via layer-wise training or better initialization strategies.

In conclusion, the paper establishes that combining variational quantum algorithms with tensor network (MPS) techniques is a robust strategy for preparing and characterizing finite-temperature quantum states on current hardware, provided that appropriate error mitigation and ansatz selection (favoring HEA for low temperatures) are employed.