Efficient quantum-enhanced classical simulation for patches of quantum landscapes

TL;DR

The paper addresses the challenge of identifying where quantum advantage is essential by introducing a framework to surrogatably simulate patches of quantum expectation landscapes. It develops a hybrid quantum-classical approach where a polynomial-time classical surrogate is built from a polynomial-time data-acquisition phase on a quantum device, enabling fully classical evaluation within a patch around a chosen point $\boldsymbol{\alpha}^*$ with width $r$. Central to the method are (i) a general surrogate guarantee for arbitrary parameterized channels and (ii) a near-Clifford focused Pauli-propagation technique that yields efficient time- and sample-scales when the patch lies near Clifford circuits. The paper substantiates these ideas with rigorous error-scalar results and numerical demonstrations on a Hamiltonian variational ansatz and a 127-qubit heavy-hex topology, revealing practical pathways to extend quantum resources where they matter most. Overall, it broadens the landscape of quantum-enhanced classical computation, offering a concrete, scalable bridge between current quantum devices and classical optimization and simulation tools.

Abstract

Understanding the capabilities of classical simulation methods is key to identifying where quantum computers are advantageous. Not only does this ensure that quantum computers are used only where necessary, but also one can potentially identify subroutines that can be offloaded onto a classical device. In this work, we show that it is always possible to generate a classical surrogate of a sub-region (dubbed a "patch") of an expectation landscape produced by a parameterized quantum circuit. That is, we provide a quantum-enhanced classical algorithm which, after simple measurements on a quantum device, allows one to classically simulate approximate expectation values of a subregion of a landscape. We provide time and sample complexity guarantees for a range of families of circuits of interest, and further numerically demonstrate our simulation algorithms on an exactly verifiable simulation of a Hamiltonian variational ansatz and long-time dynamics simulation on a 127-qubit heavy-hex topology.

Figures (4)

• Figure 1: The process of surrogating the expectation function $f(\boldsymbol{\alpha})$ obtained as the expectation value of an observable $O$, measured over a state $\rho$ evolved under the $m$-parameter circuit $U(\boldsymbol{\alpha})$. Quantum measurements and/or classical simulations via truncated Pauli propagation are used to surrogate a patch $\overline{f}(\boldsymbol{\alpha})$ which can be used in lieu of further quantum operations for applications such as VQA.
• Figure 2: A visualisation of Pauli propagation computing the expectation value of observable $O = ZX$ resulting from the pictured two-qubit circuit upon input state $\rho$. Evaluation proceeds left-to-right, simulating the circuit in reverse; graph edges indicate the Pauli strings produced by the gate aligned above them, and their labels indicate the induced coefficient; elements of the gate's Pauli transfer matrix. The Clifford CNOT gates merely multiply Pauli strings by $\pm 1$ (blue edges), while the non-Clifford rotations $R_P(\theta) = e^{-i \theta P/2 }$ can sometimes produce two Pauli strings (pink edges). Notice the state $\rho$ need not be explicitly instantiated; instead, only its trace with respect to the final Pauli strings are evaluated.
• Figure 3: Landscape and surrogation root mean square error (RMSE) as a function of classical and quantum resources. We consider a 16-qubit Hamiltonian variational ansatz of a transverse field Ising (TFI) model on a 4x4 grid and a 1-local Pauli $Z$ expectation value near the middle of the grid. The initial state $\rho$ is the all-zero state evolved exactly by 4 Trotter layers of the TFI Hamiltonian at $dt=0.1$, and we surrogate the final 4 layers with free parameters. a) Truncation RMSE of our small-angle Pauli propagation surrogate for different $r$ as a function of the number of backpropagated Pauli operators. This number corresponds to increasing sine-expansion order from $\kappa =0$ to $\kappa =10$. b) Total RMSE for expectation values using measurements of the initial state $\rho$ for different $r$ as a function of the number of quantum measurements at $\kappa =6$. We use the effective 1-norm of the surrogate's coefficients described to allocate the measurements on $\rho$. c) Comparison of different measurement allocation approaches at $\kappa =6$ and $r=0.1$. The uniform allocation over the operators expectedly underperforms and Pauli classical shadows are in this case on-par with our effective 1-norm allocation.
• Figure 4: Testing the KZ scaling of defects $n_\text{def}$ in a time-dependent TFI model with 127 spins on a heavy-hex entangling topology. Using our Pauli propagation surrogate, we verify the conjectured scaling resembling $t_f^{-1/2}$ on the heavy-hex topology. The ramp function $g(t)$ in Eq. \ref{['eq:TFI']} is varied to confirm a similar scaling across various annealing protocols. The surrogation of the expectation landscape with 50 Trotter steps and $m=13550$ gates was done in under 30 seconds, with every expectation evaluation taking under one second.

Theorems & Definitions (44)

• Theorem 1: General Surrogation Guarantee, Informal
• Theorem 2: Time complexity of small-angle Pauli propagation, Informal
• Theorem 3: Surrogation via Direct Pauli Measurements, Informal
• Theorem 4: Shadow Surrogation Guarantee, Informal
• Lemma 1: Markov's inequality
• Lemma 2: Hoeffding's inequality hoeffding1963probability
• Lemma 3
• proof
• Lemma 4
• proof