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arXiv:2301.07477 · Physical Review A

Shallow quantum circuits for efficient preparation of Slater determinants and correlated states on a quantum computer

Chong Hian Chee, Daniel Leykam, Adrian Matthew Mak, Dimitris G. Angelakis

Published Aug 18, 2023• 6 citations
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Abstract

Fermionic Ansatz state preparation is a critical subroutine in many quantum algorithms such as the variational quantum eigensolver for quantum chemistry and condensed-matter applications. The shallowest circuit depth needed to prepare Slater determinants and correlated states to date scales at least linearly with respect to the system size $N$. Inspired by data-loading circuits developed for quantum machine learning, we propose an alternate paradigm that provides shallower, yet scalable, $O(d{log}_{2}^{2}N)$ two-qubit gate-depth circuits to prepare such states with $d$ fermions, offering a subexponential reduction in $N$ over existing approaches in second quantization, enabling high-accuracy studies of $d\ensuremath{\ll}O(N/{log}_{2}^{2}N)$ fermionic systems with larger basis sets on near-term quantum devices.

DOI
10.1103/physreva.108.022416

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Fermionic Ansatz state preparation is a critical subroutine in many quantum algorithms such as the variational quantum eigensolver for quantum chemistry and condensed-matter applications. The shallowest circuit depth needed to prepare Slater determinants and correlated states to date scales at least linearly with respect to the system size NNN. Inspired by data-loading circuits developed for quantum machine learning, we propose an alternate paradigm that provides shallower, yet scalable, O(dlog22N)O(d{log}_{2}^{2}N)O(dlog22​N) two-qubit gate-depth circuits to prepare such states with ddd fermions, offering a subexponential reduction in NNN over existing approaches in second quantization, enabling high-accuracy studies of d\ensuremath≪O(N/log22N)d\ensuremath{\ll}O(N/{log}_{2}^{2}N)d\ensuremath≪O(N/log22​N) fermionic systems with larger basis sets on near-term quantum devices.
Abstract Quantum simulations are bound to be one of the main applications of near-term quantum computers. Quantum chemistry and condensed matter physics are expected to benefit from these technological developments. Several quantum simulation methods are known to prepare a state on a quantum computer and measure the desired observables. The most resource economic procedure is the variational quantum eigensolver (VQE), which has traditionally employed unitary coupled cluster as the ansatz to approximate ground states of many-body fermionic Hamiltonians. A significant caveat of the method is that the initial state of the procedure is a single reference product state from a classical Hartree–Fock calculation with no pairing correlations, hence it cannot represent superconducting states. In this work, we propose to improve the method by initializing the algorithm with a more general fermionic Gaussian state, an idea borrowed from the field of nuclear physics. We show how this Gaussian reference state can be prepared with a linear-depth circuit of quantum matchgates. By augmenting the set of available gates with nearest-neighbor phase coupling, we generate a low-depth circuit ansatz that can accurately prepare the ground state of correlated fermionic systems. This extends the range of applicability of the VQE to systems with strong pairing correlations such as superconductors, atomic nuclei, and topological materials.
Simulating computationally hard fermionic systems is a promising application of quantum computing. However, mapping nonlocal fermionic operators to qubits often produces deep circuits, rendering such simulations impractical on near-term hardware. We introduce two Majorana swap network compilation strategies for variational quantum eigensolvers that reduce circuit depth and two-qubit gate count. First, we develop a cyclic compilation algorithm that localizes all two-particle interaction terms in a general fermionic Hamiltonian containing up to O(M4)\mathcal{O}(M^4)O(M4) such terms using only O(M3)\mathcal{O}(M^3)O(M3) auxiliary Majorana-swap transpositions, where MMM is the number of fermionic modes. Here, the cubic scaling refers to auxiliary routing; a complete UCCGSD ansatz still contains O(M4)\mathcal{O}(M^4)O(M4) double-excitation rotations. Second, we design a Majorana swap network for the kkk-UpCCGSD variational ansatz, which is already more compact than UCCGSD. In this setting, our network yields constant-factor reductions of approximately 505050 % in circuit depth and 202020 % in two-qubit gate count under all-to-all connectivity. For the more restricted 2×N2\times N2×N connectivity, the reductions are larger --- about 555555 % in circuit depth an 404040 % in gate count. These structural improvements are accompanied by improved robustness in numerical noise simulations on the small molecular instances tested.
Quantum algorithms require accurate representations of electronic states on a quantum device, yet the approximation of electronic wave functions for strongly correlated systems remains a profound theoretical challenge, with existing methods struggling to balance the competing demands of chemical accuracy and gate efficiency. Moreover, a critical limitation of the most of the state-of-the-art methods developed to date lies in their substantial reliance on extensive pre-circuit measurements, which introduce significant overheads and contribute to inefficiencies in practical implementation. To address these interconnected challenges and establish a harmonious synergy between them, we propose an algorithmic framework that focuses on efficiently capturing the molecular strong correlation through an ordered set of computationally less demanding rank-one and seniority-zero excitations, yielding a parameterized ansatz with shallow gate depth. Furthermore, to achieve minimal pre-circuit measurement overhead, we implement a selective pruning of excitations through a hybrid approach that combines intuition-based selection with shallow-depth, rank-one excitations driven uni-parameter circuit optimization strategy. With the incorporation of qubit-based excitations via particle-preserving exchange circuits, we demonstrate a further reduction in quantum complexities, enhancing the overall resource efficiency of the approach. With a range of challenging applications on strongly correlated systems, we demonstrate that our dynamic ansatz not only significantly enhances computational efficiency but also delivers exceptional accuracy, robustness, and resilience to the noisy environments inherent in near-term quantum hardware.
Abstract Efficient computation of molecular energies is an exciting application of quantum computing for quantum chemistry, but current noisy intermediate-scale quantum (NISQ) devices can only execute shallow circuits, limiting existing variational quantum algorithms, which require deep entangling quantum circuit ansatzes to capture correlations, to small molecules. Here we demonstrate a variational NISQ-friendly algorithm that generates a set of Hartree–Fock ansatzes using multiple shallow circuits with depth linear in the number of qubits to estimate electronic correlation energies via perturbation theory up to the second order. We tested the algorithm on several small molecules, both with classical simulations including noise models and on cloud quantum processors, showing that it not only reproduces the equilibrium molecular energies but it also captures the perturbative electronic correlation effects at longer bond distances. The number of shallow circuits required for our algorithms scales O ( N 5 ) , with N being the system size, thus enabling the study of larger molecules compared to other approaches, where circuit depth or number of qubits required become prohibitive for NISQ devices.
Data loading with shallow quantum circuits is a highly desirable ingredient for efficient execution of many quantum algorithms before large-scale quantum error corrections with full fault tolerance become readily available. The need for efficient data loading is especially urgent for the study of quantum machine learning. In this work, we propose a protocol that only uses a parameterized shallow quantum circuit without ancilla qubits for loading data into the amplitude of a quantum state with high fidelity. We term this data-loading method Hamiltonian-based data loading (HDL), which comprises two stages. First, the HDL algorithm identifies a Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ whose unique ground state $|\ensuremath{\psi}\ensuremath{\rangle}$ represents the normalized data $\stackrel{P\vec}{x}$ in the form of amplitude encoding. Next, the target state is reconstructed with a parameterized quantum circuit by utilizing methods like variational quantum eigensolver to minimize energy. In this work, we provide three convincing examples to demonstrate the effectiveness of HDL for loading an NNN-dimensional classical data with minimal quantum resources (O[poly(log2N)]O[\mathrm{poly}({log}_{2}N)]O[poly(log2​N)]-depth quantum circuit without ancilla qubits). Our approach is particularly useful for quantum hardware without large-scale error corrections, and shall benefit the development of quantum machine learning algorithms.
Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids of up to six sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver (VQE) called ADAPT-VQE [H. R. Grimsley et al., Nat. Commun. 10, 3007 (2019)]. In contrast with nonadaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the nonadaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green's functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach [A. Peruzzo et al., Nat. Commun. 5, 4213 (2014); J. R. McLean et al., New J. Phys. 18, 023023 (2016); K. Sharma et al., New J. Phys. 22, 043006 (2020)], and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well suited for noisy intermediate-scale quantum devices.