[Paper Review] Generation of Thermofield Double States and Critical Ground States with a Quantum Computer

Daiwei Zhu, Sonika Johri+7 more|arXiv (Cornell University)|Jun 6, 2019

Quantum and electron transport phenomena46 references112 citations

TL;DR

This paper demonstrates the experimental preparation of thermofield double (TFD) states and critical ground states of the transverse-field Ising model on a trapped-ion quantum computer using a variational quantum algorithm inspired by QAOA. By applying a minimal-depth circuit with tunable evolution operators, the authors achieve high fidelity TFD states at various temperatures—including zero temperature—enabling simulation of entangled thermal states relevant to black hole physics and quantum criticality.

ABSTRACT

Finite-temperature phases of many-body quantum systems are fundamental to phenomena ranging from condensed-matter physics to cosmology, yet they are generally difficult to simulate. Using an ion trap quantum computer and protocols motivated by the Quantum Approximate Optimization Algorithm (QAOA), we generate nontrivial thermal quantum states of the transverse-field Ising model (TFIM) by preparing thermofield double states at a variety of temperatures. We also prepare the critical state of the TFIM at zero temperature using quantum-classical hybrid optimization. The entanglement structure of thermofield double and critical states plays a key role in the study of black holes, and our work simulates such nontrivial structures on a quantum computer. Moreover, we find that the variational quantum circuits exhibit noise thresholds above which the lowest depth QAOA circuits provide the best results.

Motivation & Objective

• To simulate finite-temperature many-body quantum states, particularly thermofield double (TFD) states, which are pure entangled states dual to black hole geometries in holography.
• To prepare the critical ground state of the transverse-field Ising model at zero temperature using hybrid quantum-classical optimization.
• To demonstrate that near-term quantum devices can simulate complex entanglement structures associated with quantum criticality and thermalization.
• To develop and validate symmetry-based error mitigation to improve fidelity in noisy intermediate-scale quantum (NISQ) devices.

Proposed method

• Uses a variational quantum circuit based on the Quantum Approximate Optimization Algorithm (QAOA) to prepare TFD states by alternating between inter-system (HAB) and intra-system (HA + HB) evolution operators.
• Employs a minimal ansatz with four evolution layers: HABZ, HABX, HXX, and HZ, tailored to the 6-qubit trapped-ion platform.
• Initial state is a product of three Bell-pair singlets, representing an infinite-temperature TFD, which is then unitarily evolved to target finite-temperature states.
• Optimizes variational parameters (α1, α2, γ1, γ2) classically to maximize fidelity with the target TFD state.
• Applies Z2 symmetry-based error mitigation by post-selecting only measurement outcomes that preserve the Z2 symmetry (i.e., ∏Zi,A = −∏Zi,B).
• Uses quantum-classical feedback to prepare the zero-temperature critical ground state of the TFIM with seven ions.

Experimental results

Research questions

• RQ1 Can thermofield double states of the transverse-field Ising model be prepared on a near-term quantum computer using variational circuits?
• RQ2 What is the performance of a minimal-depth variational circuit in preparing TFD states across a range of temperatures, including the critical point?
• RQ3 How does symmetry-based error mitigation improve the fidelity of TFD state preparation in noisy quantum hardware?
• RQ4 Can the critical ground state of the TFIM be prepared directly using hybrid quantum-classical optimization on a trapped-ion platform?
• RQ5 What is the noise threshold behavior of the variational circuits, and at what point do shallow circuits outperform deeper ones?

Key findings

• The protocol achieves TFD state fidelities of 0.93 at zero temperature and up to 1.0 at infinite temperature, demonstrating high-fidelity state preparation with minimal circuit depth.
• For a 3-qubit ring system (6 qubits total), a single-layer circuit (p=1) suffices to perfectly prepare the target TFD state at any temperature.
• Symmetry-based error mitigation significantly improves the accuracy of cross-system correlators, with measurable improvement in both TFD and critical ground state measurements.
• The selection rate for symmetry post-selection decreases with increasing temperature, consistent with higher error rates at elevated temperatures.
• The study identifies a noise threshold above which the lowest-depth QAOA circuits yield the best results, indicating optimal performance in noisy regimes.
• The critical ground state of the TFIM is successfully prepared using quantum-classical feedback, confirming the feasibility of direct preparation of zero-temperature critical states.