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Auxiliary-Field Quantum Monte Carlo on Quantum Hardware via Unitary Dilation

Xiantao Li

Abstract

We present near-term quantum algorithms for auxiliary-field quantum Monte Carlo (AFQMC), which represents imaginary-time projection for ground-state calculation as an ensemble of one-body propagators driven by stochastic fields $Ω$. Starting from the Feynman-Kac formula, we convert each trajectory into a sequence of piecewise-constant one-body generators using stochastic Magnus expansions up to second order, and embed the resulting nonunitary slices into unitaries with a small ancilla overhead. This lifts the projector dynamics to a unitary evolution, enabling coherent circuit execution in the regime $\|Ω\| τ=O(1)$ and reducing the need for frequent mid-circuit measurement. We further derive an equivalent linear-combination-of-unitaries (LCU) form that yields system-only, shallower circuits by trading ancilla cost for additional trajectory sampling. Benchmarks on the Hubbard model verify the accuracy of the dilation and Magnus expansions classically and demonstrate multi-step executions on IBM quantum hardware.

Auxiliary-Field Quantum Monte Carlo on Quantum Hardware via Unitary Dilation

Abstract

We present near-term quantum algorithms for auxiliary-field quantum Monte Carlo (AFQMC), which represents imaginary-time projection for ground-state calculation as an ensemble of one-body propagators driven by stochastic fields . Starting from the Feynman-Kac formula, we convert each trajectory into a sequence of piecewise-constant one-body generators using stochastic Magnus expansions up to second order, and embed the resulting nonunitary slices into unitaries with a small ancilla overhead. This lifts the projector dynamics to a unitary evolution, enabling coherent circuit execution in the regime and reducing the need for frequent mid-circuit measurement. We further derive an equivalent linear-combination-of-unitaries (LCU) form that yields system-only, shallower circuits by trading ancilla cost for additional trajectory sampling. Benchmarks on the Hubbard model verify the accuracy of the dilation and Magnus expansions classically and demonstrate multi-step executions on IBM quantum hardware.
Paper Structure (21 sections, 3 theorems, 92 equations, 4 figures)

This paper contains 21 sections, 3 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods and Algorithms
  3. Ground-state projection and AFQMC structure
  4. Feynman--Kac representation and stochastic differential equations (SDEs)
  5. Stochastic Magnus expansion
  6. Weak order-1 ME.
  7. Weak order-2 ME.
  8. Unitary dilation of stochastic generators
  9. Relation to prior fault-tolerant algorithms.
  10. System-only implementation via LCU
  11. Numerical test
  12. Summary and Discussion
  13. The moment-fulfilling dilation method
  14. The Derivation of the operator bound for HS/Magnus slices
  15. LCU coefficient normalization and constant success
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\hat{H}$ be Hermitian with ground energy $E_0$ and gap $\Delta>0$, and let $\ket{\psi_T}$ have nonzero ground overlap $\gamma>0$. Choose $\tau=\Theta(1/\norm{\Omega})$ with $\tau=n_T\Delta t$ and ${\Delta t}=\Theta(\sqrt{\epsilon_{\rm prep}})$. If the per-segment normalized preparation error is then after $S=\Theta\!(\frac{\norm{\Omega}}{\Delta}\log\frac{1}{\gamma\epsilon})$ segments one obta

Figures (4)

  • Figure 1: One-step LCU kernel error $\|B_{\mathrm{LCU}}(\Omega)-e^{\Omega}\|_2$ versus ancilla size $n_A$, summarized over 100 sampled weak-order-2 Magnus slice exponents $\Omega_k$. The median and 90th percentile both decrease rapidly with $n_A$, showing that a modest ancilla is sufficient to make the one-step LCU approximation accurate in the constant-success regime.
  • Figure 2: Energy estimates versus imaginary time $T$ comparing Magnus--1 and Magnus--2 AFQMC propagation. Dashed and dash--dot curves with markers show mixed and true (Rayleigh) estimators with jackknife error bars, and solid lines denote the corresponding $\exp(-\tau\hat{H})$ references.
  • Figure 3: Energy estimates versus imaginary time $\tau$ for the two-segment Magnus--2 dilation simulation. Circles (squares) show the mixed (true/Rayleigh) estimator with path-sampling error bars; dashed (dotted) curves denote the corresponding deterministic references computed from $\exp(-\tau\hat{H})$. The vertical dash--dot line marks the segment boundary.
  • Figure 4: Mixed-estimator energies versus the number of Magnus steps $n_T$. Blue squares denote the exact mixed estimator $\langle \psi_T|H e^{-\tau H}|\psi_T\rangle / \langle \psi_T|e^{-\tau H}|\psi_T\rangle$, orange circles the statevector emulation of the compiled LCU circuits, and red stars the IBM-Boston hardware results. Error bars show bootstrap standard errors for the emulator and hardware data.

Theorems & Definitions (4)

  • Theorem 1: Fault-tolerant segmented imaginary-time projection
  • Lemma 2: $\ell_1$-bound for the LCU coefficients
  • proof
  • Theorem 3