Shallow circuits could locate quantum critical points without precise ground states

Ching-Yu Huang of Tunghai University and Min-Fong Yang have demonstrated a method for pinpointing quantum critical points using surprisingly shallow quantum circuits. Their work addresses the challenge of mapping quantum phase diagrams, a task often limited by the prohibitive computational demands of classical simulation. The researchers found that a technique called Delta-VQE becomes more effective at identifying these transitions as circuit depth decreases, suggesting a path toward resource-conscious quantum probes of quantum criticality. However, the study reveals that Delta-VQE’s success hinges on carefully tailoring circuits to reflect the specific phases being investigated.

Delta-VQE Locates Critical Points Using Energy Differences

Delta-VQE, a new quantum algorithm, prioritizes contrasting phases over precise ground-state energy calculations. The method retains the lowest-energy solution from multiple optimization runs, improving the continuity of the variational solution and reducing the chance of converging on local minima. This strategy, detailed in Section II. 2 of the research, defines the cost function as an energy functional and selects optimized energies according to Equation (2).

Fidelity susceptibility, calculated through exact diagonalization, served as a benchmark for assessing Delta-VQE’s performance, providing a well-established method for identifying quantum phase transitions. While fidelity susceptibility relies on accurate ground state calculations and requires larger system sizes to resolve critical signatures, it offers a comparative standard for evaluating the new algorithm’s efficiency. Delta-VQE’s diagnostic capability sharpens as circuit depth decreases, a counterintuitive result given the expectation that deeper circuits offer greater power.

However, the study reveals a critical limitation: when employing dual variational ansätze, Delta-VQE invariably detects the self-dual points rather than the true criticality. This occurs across all circuit depths and system sizes, meaning the algorithm misidentifies the transition point if the genuine critical point does not coincide with the self-dual point. This finding highlights the importance of carefully tailoring the variational circuits to reflect the competing phases across the boundary.

To overcome this pitfall, the researchers demonstrated that carefully tailored ansätze, embodying the competing phases, successfully identify the genuine Ising critical point with minimal finite-size effects. The core principle of Delta-VQE is a conceptual shift in using the variational quantum eigensolver for phase transition detection, emphasizing qualitative contrast between phases over the quantitative exactness of ground-state energy evaluation. This approach, validated using the generalized cluster-Ising model, offers a resource-efficient alternative to traditional methods.

Variational Quantum Eigensolver in the NISQ Era

The team’s work provides a resource-conscious approach to probing quantum criticality, potentially making it more feasible with current noisy intermediate-scale quantum (NISQ) devices. The simulations used systems of up to 12 qubits, demonstrating the method’s applicability to systems beyond the smallest scales. The findings have implications for the broader development of variational quantum algorithms, emphasizing the need for careful consideration of ansatz design and its impact on the accuracy of results.

While VQE offers a promising alternative to traditional methods for studying quantum phase transitions, its success hinges on a deep understanding of the underlying physics and the ability to translate that understanding into appropriate circuit architectures. The simulations provide a valuable benchmark for future research in this area.

Shallow Circuit Depth Enhances Delta-VQE Efficiency

Researchers employed a software development kit to construct the parameterized quantum circuits and evaluate expectation values using an ideal statevector simulator, avoiding the complications of noise in current quantum devices. Variational parameters were optimized using the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm with bound constraints, a method implemented within the SciPy optimization library, to refine the accuracy of the energy calculations. The researchers acknowledge that Delta-VQE is fundamentally a heuristic method, and a rigorous proof of its broader applicability remains an open question.

However, their work provides a foundation for developing a more general criterion for understanding and extending the method’s reach. The team’s approach involved carefully selecting pairs of ansätze, or trial wavefunctions, tailored to the specific quantum phases being compared, to accurately locate the transitions between them.

A key observation is that the use of dual ansätze invariably leads to the detection of self-dual points rather than the true critical point, unless the ansätze are specifically designed to embody the competing phases across the boundary. This limitation highlights the importance of circuit tailoring; simply applying a standard Delta-VQE protocol does not guarantee accurate results. For instance, when examining a system with an Ising coupling of zero, the minimum of the energy difference was precisely located at the expected critical point of one, even as the circuit depth was increased.

The team also investigated the impact of system size, finding that the estimated critical point systematically approached the exact value as the number of qubits increased, indicating diminishing finite-size corrections. “As discussed at the end of Sec. II.3, the states and constitute a dual pair,” the researchers write, explaining the underlying physics driving the accuracy of their approach. This pairing of states is essential for effectively contrasting the energies of the competing phases and identifying the transition point.

Phase Transition Detection Beyond Ground State Precision

The team’s work builds on the premise that faithfully preparing the exact ground state is often unnecessary for constructing phase diagrams, with low-fidelity approximations proving sufficient to reveal qualitative features and locate critical points. The choice of reference states within Delta-VQE also proved critical to achieving accurate results. For the ferromagnetic phase, the symmetry-preserving Greenberger-Horne-Zeilinger (GHZ) state was used, rather than symmetry-breaking product states, because “spontaneous symmetry breaking does not occur in a finite system.” This selection yielded improved quantitative accuracy in identifying the transition.

Similarly, for the cluster phase, the reference state was defined as the cluster state, corresponding to the ground state of -H_(XZX). Shi et al. of the Delta-VQE algorithm. The team’s analysis of the energy difference, Δ E(h_c,J) = |E_1(h_c,J)-E_2(h_c,J)|, revealed that even when optimized states deviate from the exact ground state, the energy difference can remain small if the spectral weights are similar for all low-lying levels.

This equation highlights the method’s reliance on accurately capturing the energy differences between competing phases. (3) can be expressed as: Δ E(h_c,J) = sum_(n > 0)(|$a_n$|^2-|$b_n$|^2)$\Delta {\epsilon }_n$, where denotes the excitation energy with ${\epsilon }_n$ being the eigenenergy associated with the state |n⟩.” The exact critical points were reproduced even with a single-layer ansatz and for small system sizes in the first three models, further solidifying the method’s potential as a robust tool for quantum phase transition detection.

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