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Scaling of variational quantum circuit depth for condensed matter systems

Bravo Prieto, Carlos, Lumbreras Zarapico, Josep, Luca Tagliacozzo, José Ignacio Latorre

Published May 20, 2020• 51 citations
OpenAlex

Abstract

We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A finite-depth regime, where the accuracy improves slowly with the number of layers, and a finite-size regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases linearly with the size of the system. We discuss the implication of these observations in the context of comparing different variational ansatz and their effectiveness in describing critical ground states.

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We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A finite-depth regime, where the accuracy improves slowly with the number of layers, and a finite-size regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases linearly with the size of the system. We discuss the implication of these observations in the context of comparing different variational ansatz and their effectiveness in describing critical ground states.
We benchmark the accuracy of a variational quantum eigensolver based on a finite-depth quantum circuit encoding ground state of local Hamiltonians. We show that in gapped phases, the accuracy improves exponentially with the depth of the circuit. When trying to encode the ground state of conformally invariant Hamiltonians, we observe two regimes. A finite-depth regime, where the accuracy improves slowly with the number of layers, and a finite-size regime where it improves again exponentially. The cross-over between the two regimes happens at a critical number of layers whose value increases linearly with the size of the system. We discuss the implication of these observations in the context of comparing different variational ansatz and their effectiveness in describing critical ground states.
We propose an efficient circuit structure of variational quantum circuit Ans\"atze used for the variational quantum eigensolver (VQE) algorithm in calculating gapped topological phases on the currently feasible noisy intermediate-scale quantum computers. An efficient circuit Ansatz should include two layers: the initialization layer and the variational layer. In the initialization layer, a fixed-depth circuit state with a compatible entanglement structure to the target topological phase is constructed. The circuit state is further adjusted subsequently to capture the details of the local correlations, which is dictated with the Hamiltonian, in the parametrized variational layer. Based on this strategy, we design a circuit Ansatz to investigate the symmetry-protected topological Haldane phase in a nonexactly solvable alternating spin-12\frac{1}{2}21​ Heisenberg chain by VQE calculations. Main characterizations of the Haldane phase, including the long-ranged string order, the fourfold nearly degenerate ground states associated with four different localized edge-mode patterns for the system with open boundaries, and the twofold degeneracy of the entanglement spectrum, are all observed for the optimized shallow circuit state with only one-depth variational layer both in numerical simulations and on real quantum computers. We further demonstrate that the computational capacity (i.e., expressibility) of this quantum circuit Ansatz is determined not by the system size but only by the intrinsic correlation length of the system, thus implying that the scalable VQE calculation is possible.
The scaling of the entanglement entropy at a quantum critical point allows us to extract universal properties of the state, e.g., the central charge of a conformal field theory. With the rapid improvement of noisy intermediate-scale quantum (NISQ) devices, these quantum computers present themselves as a powerful tool to study critical many-body systems. We use finite-depth quantum circuits suitable for NISQ devices as a variational ansatz to represent ground states of critical, infinite systems. We find universal finite-depth scaling relations for these circuits and verify them numerically at two different critical points, i.e., the critical Ising model with an additional symmetry-preserving term and the critical XXZ model.
A variational quantum eigensolver (VQE) solves the ground state problem of a given Hamiltonian by finding the parameters of a quantum circuit Ansatz that minimizes the Hamiltonian expectation value. Among possible quantum circuit Ansätze, the Hamiltonian variational Ansatz (HVA) is widely studied for quantum many-body problems as the Ansatz with sufficiently large depth is theoretically guaranteed to express the ground state. However, since the HVA shares the same symmetry with the Hamiltonian, it is not necessarily good at finding symmetry-broken ground states that prevail in nature. In this paper, we systematically explore the limitations of the HVA for solving symmetry-broken systems and propose an alternative quantum circuit Ansatz with symmetry-breaking layers. With extensive numerical simulations, we show that the proposed Ansatz finds the ground state in depth significantly shorter than the bare HVA when the target Hamiltonian has symmetry-broken ground states.
Ground state preparation is classically intractable for general Hamiltonians. On quantum devices, shallow parametrized circuits can be effectively trained to obtain short-range entangled states under the paradigm of variational quantum eigensolver, while deep circuits are generally untrainable due to the barren plateau phenomenon. In this Letter, we give a general lower bound on the variance of circuit gradients for arbitrary quantum circuits composed of local 2-designs. Based on our unified framework, we prove the absence of barren plateaus in training finite local-depth circuits (FLDC) for the ground states of local Hamiltonians. FLDCs are allowed to be deep in the conventional circuit depth to generate long-range entangled ground states, such as topologically ordered states, but their local depths are finite, i.e., there is only a finite number of gates acting on individual qubits. This characteristic sets FLDC apart from shallow circuits: FLDC in general cannot be classically simulated to estimate local observables efficiently by existing tensor network methods in two and higher dimensions. We validate our analytical results with extensive numerical simulations and demonstrate the effectiveness of variational training using the generalized toric code model.
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H 4 and the π -system of hexatriene (C 6 H 8 ).