Quantum computing quantum Monte Carlo algorithm

TL;DR

The paper addresses the sign problem and circuit-depth limitations that hinder solving ground-state problems in strongly correlated quantum systems. It introduces QC-FCIQMC, a hybrid quantum-classical algorithm that replaces classical Slater-determinant walkers with quantum walkers |\phi_i\rangle=U|i\rangle, where U is prepared by a shallow VQE circuit to yield a Walker basis closer to the ground state. Non-stoquasticity indicators (NSIs) with computable upper bounds quantify the sign problem and guide basis rotations to mitigate it, while quantum circuits efficiently sample Hamiltonian couplings |\widetilde{H_{ij}}| and implement spawning via a Bernoulli factory, enabling scalable imaginary-time evolution without exponential resource costs. Numerical tests on N2 (12 qubits) and the 2×4 Hubbard model (16 qubits) show substantial reductions in NSIs and energy variance, and orders-of-magnitude fewer walkers are needed as the VQE-based basis deepens, achieving chemical accuracy with shallow circuits. Overall, the work demonstrates a practical pathway for near-term quantum hardware to tackle realistic quantum chemistry and condensed-matter problems, supported by NSI bounds that quantify and guide sign-problem mitigation.

Abstract

Quantum computing and quantum Monte Carlo (QMC) are respectively the state-of-the-art quantum and classical computing methods for understanding many-body quantum systems. Here, we propose a hybrid quantum-classical algorithm that integrates these two methods, inheriting their distinct features in efficient representation and manipulation of quantum states and overcoming their limitations. We first introduce non-stoquasticity indicators (NSIs) and their upper bounds, which measure the sign problem, the most notable limitation of QMC. We show that our algorithm could greatly mitigate the sign problem, which decreases NSIs with the assistance of quantum computing. Meanwhile, the use of quantum Monte Carlo also increases the expressivity of shallow quantum circuits, allowing more accurate computation that is conventionally achievable only with much deeper circuits. We numerically test and verify the method for the N$_2$ molecule (12 qubits) and the Hubbard model (16 qubits). Our work paves the way to solving practical problems with intermediate-scale and early-fault tolerant quantum computers, with potential applications in chemistry, condensed matter physics, materials, high energy physics, etc.

Figures (7)

• Figure 1: (a) Sketch of the spawning, death/cloning, and annihilation steps in FCIQMC. Red and blue colors represent walkers with positive and negative signs respectively. Red and blue arrows represent the sign of $H_{ji}$. The transparent boxes represent dead walkers, while the elongated walkers are the cloned ones. (b) A sketched cartoon of the expected sign problem comparison for our QC-FCIQMC and FCIQMC. (c) The procedure of QC-FCIQMC. The quantum circuit $U$ that generates the new basis is obtained from VQAs, such as the one shown in (e). (d) A sketched cartoon of the expected energy convergence comparison of ADAPT-VQE and QC-FCIQMC. (f) Key quantum circuits for realizing walker propagation. Here $\ket{\phi_i} = U\ket{i}$ is the quantum state walker with the unitary $U$, the controlled-$P_{(k,k')}$ gate denotes $\ket{0}\bra{0}\otimes P_k + \ket{1}\bra{1}\otimes P_{k'}$ with $P_k$ being Pauli matrices. When the Hamiltonian has the decomposition $H=\sum_k h_kP_k$, the quantum circuit evaluates/samples according to $p_{k,k'}^i(j)=\mathrm{Re}\langle{i|U^\dag P_k U\Pi_j U^\dag P_{k'}U|i}\rangle$, which constitute $\abs{\widetilde{H_{ji}}}^2=\sum_{k,k}h_kh_{k'}p_{k,k'}^i(j)$, here $\Pi_j=\ket{j}\bra{j}$.We refer to the main text for the protocol detailing how to use the quantum circuit to achieve the propagation of quantum state walkers. (g) Circuit for evaluating the real part of $\widetilde{H_{ji}}$. The imaginary part could be similarly obtained by adding an $S$ gate before the last $H$ gate. The circuit is used to track the sign/phase during walker propagation.
• Figure 2: Numerical results of ADAPT-VQE, FCIQMC, and QC-FCIQMC for the N$_2$ molecule with 12 qubits. (a) Potential energy surface for the N$_2$ molecule with different methods under the STO-3g basis set. (b) The error and standard deviation of the average energy for different bond lengths. Here the quantum state walkers used in QC-FCIQMC are prepared with quantum circuits from ADAPT-VQE (adding 12 fermionic operators for all bond lengths). (c) Comparison of the average energy and standard deviation for ADAPT-VQE and QC-FCIQMC with different circuit depths and a bounded number of walkers. (d) Standard deviations from (c) as well as the bounds for non-stoquastic indicator (NSI) with $\beta=10^{-1}$. (e) Log 10 plots of the numbers of walkers needed to achieve a certain precision for the energies, as well as the NSI values for reference. (f) QC-FCIQMC on N$_2$ molecule at bond length 4.0$\AA$ with sampling noise. Here we use the VQE basis generated by 12 layers of ADAPT-VQE and the $x$-axis denotes the total number of samples of the whole QC-FCIQMC process.
• Figure 3: Numerical results of QC-FCIQMC for the 2x4 Hubbard model ($U/t=4$) with 16 qubits. The Hamiltonian variational ansatzs with 1, 5, 10, and 15 layers are implemented for comparison. The energy shift starts at $10^4$ walkers for all cases. (a) Comparison of energy fluctuation for FCIQMC and QC-FCIQMC. (b)-(e) Walker population for single determinant FCIQMC, QC-FCIQMC with HV ansatz of 1 layer, 5 layers, 10 layers respectively. (f) Energy estimation for different setups of layer number. (g) Effect of variance suppression and the NSI values.
• Figure 4: (a) Circuit for estimating amplitudes $\text{Re}(\bra{\phi_j}H\ket{\phi_i})$. One let $V=I$ when estimating the real parts and $V=S^{\dagger}=100-i$ for the imaginary parts. The gate $X_{i,j}$ implements $X_i$ if the control register is $\ket{0}$ and $X_j$ if the control register is $\ket{1}$. $X_i$ implements $X$ gates on those qubits where the corresponding digits in $i$ are 1. (b) Circuit for sampling the significant amplitudes in the distribution $|\widetilde{H_{ji}}|^2$ for a fixed $i$. The gate $P_{k,k'}$ implements $P_k$ if the control register is $\ket{0}$ and $P_{k'}$ if the control register is $\ket{1}$.
• Figure 5: (a) Potential energy surface for the nitrogen molecule with QC-FCIQMC under the STO-3g basis set. (b) The standard deviation of energy evaluations along the QMC evolution. Here the new walker space of QC-FCIQMC is prepared with circuits from ADAPT-VQE (adding 12 fermionic operators for all bond lengths). Even with the assistance of shallow-depth ADAPT-VQE runs, QC-FCIQMC manages to reach chemical accuracy across all bond lengths using only around 10000 walkers. We also include data from FCIQMC with single-determinant walkers for comparison. For both single determinant FCIQMC and QC-FCIQMC, we start the energy shift when the total number of walkers exceeds 10000. Under this setting, single determinant FCIQMC fails to reach chemical accuracy at several bond lengths towards the dissociation limit. Moreover, the standard deviation from the energy profiling of QC-FCIQMC is smaller than the single determinant FCIQMC for all bond lengths. In fact, one observes a bigger variance reduction with a stronger static correlation. (c) FCIQMC-assisted ADAPT-VQE results for nitrogen molecule at bond length 4.0. ADAPT-VQE energies and QC-FCIQMC energies with standard deviations for different depths of ADAPT-VQE. (d) Log plot with standard deviations from (c) as well as the non-stoquastic indicator, where choose $\beta=10^{-1}$. FCIQMC is able to push ADAPT-VQE results of different depths to chemical accuracy. Here we obtain the expected energy by taking the average of all FCIQMC energies after it converges. We fix the total walker number at the same level by starting to implement an energy shift of $S$ when the total walker number exceeds $10^4$. The deeper the ADAPT-VQE circuit is, the smaller the energy variance FCIQMC would have. One can see with an ADAPT-VQE optimization that picks 20 operators, the energy variance obtained is already at a similar level as chemical accuracy requirement and with 24 operators the variance lies well within chemical accuracy area. (e) ADAPT-VQE energies and QC-FCIQMC energies with variances for different depths of ADAPT-VQE. (f) Log 10 plots of the respective numbers of walkers that give the energies in (e) as well as the natural log of the non-stoquastic indicator, where choose $\beta=10^{-1}$. To reach a certain level of energy variance, FCIQMC requires fewer walkers assisted with deeper ADAPT-VQE circuits. Here we choose the energy variance upper bound to be 5mHa for the number of walkers to be friendly to the running time of our code. To reach this precision, QC-FCIQMC requires about 100 times fewer walkers going from ADAPT-VQE with 4 operators to that of 24 operators.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Lemma 3: Sample complexity for estimating $\widetilde{H_{ji}}$