Improved Quantum Computation using Operator Backpropagation

TL;DR

This work tackles the limitation of decoherence in near-term quantum hardware by introducing Operator Backpropagation (OBP), a hybrid quantum-classical approach that splits a quantum circuit into a classical Heisenberg-backpropagated observable and a quantum Schrödinger-evolved subcircuit. The backpropagated operator $O' = U_C^ abla O U_C$ is decomposed into Pauli strings and measured via the shallower quantum circuit, trading quantum-depth for classical computation and additional circuit executions. The CPT-based OBP method includes truncation budgets across slices, with error bounds derived from both $L_1$ and $L_2$ norms and per-slice budgeting, and can be parallelized using ZX-calculus addressing for distributed Pauli-term deduplication. Experiments on 75- and 127-qubit XY models validate OBP's ability to reduce error in Hamiltonian-time dynamics versus purely quantum runs, illustrating potential for deeper, more accurate simulations on noisy devices and offering a path to extending near-term quantum capabilities. Overall, OBP provides a practical route to higher-precision quantum simulations by balancing depth reduction against classical overhead and measurement commerce, with broad implications for quantum chemistry and condensed-matter physics simulations.

Abstract

Decoherence of quantum hardware is currently limiting its practical applications. At the same time, classical algorithms for simulating quantum circuits have progressed substantially. Here, we demonstrate a hybrid framework that integrates classical simulations with quantum hardware to improve the computation of an observable's expectation value by reducing the quantum circuit depth. In this framework, a quantum circuit is partitioned into two subcircuits: one that describes the backpropagated Heisenberg evolution of an observable, executed on a classical computer, while the other is a Schrödinger evolution run on quantum processors. The overall effect is to reduce the depths of the circuits executed on quantum devices, trading this with classical overhead and an increased number of circuit executions. We demonstrate the effectiveness of this method on a Hamiltonian simulation problem, achieving more accurate expectation value estimates compared to using quantum hardware alone.

Figures (10)

• Figure 1: Operator backpropagation (OBP) framework. A quantum circuit $U$ is split into two subcircuits $U_C$ and $U_Q$. A classical simulator computes the Pauli decomposition of $O' = U_C^\dag O U_C$, which is then measured on quantum hardware.
• Figure 2: Comparison between $L_1$ and $L_2$ error bounds vs. exact error of OBP truncation. The exact error $\norm{\Delta}$ from truncating a backpropagated observable (blue circles) and the error bounds based on the $L_1$ norm [\ref{['eq:l1-norm-bound']}, orange dashed line] and the $L_2$ norm [\ref{['eq:l2-norm-bound']}, blue dotted line] of the truncated coeffcients. Here, $U_Q$ and $U_C$ each correspond to 5 Trotter step time evolution circuits for a 12 qubit XY model [\ref{['eq:Trotterized_evolution']}] on a one-dimensional lattice with closed boundary conditions at Trotter step size $dt = 0.1$. The initial state is $\ket{00\dots0}$ and the target observable is $Z_1$. The total number of Pauli terms in the backpropagated observable without truncation is 271.
• Figure 3: Example of distributing Pauli terms based on their ZX calculus index. From left to right we show how a Pauli term gets encoded in the ZX calculus which then gets interpreted as an address that can be mapped into the address range associated with a given node.
• Figure 4: OBP experiments with 75 and 127-qubit spin models. Benchmarking the OBP framework in the simulation of the one-dimensional XY model of 75 spins [panels a) and b)] and the two-dimensional XY model of 127 spins [panels c) and d)]. a/c) Expectation of the polarization $M$ at different time steps. The polarization is a conserved quantity (dashed line) under the dynamics of the XY model. Due to noise, the experimental signal decays with the depth of the circuit and can be partially recovered using error mitigation. The signals at different noise amplification (by a factor of 1, 1.5, 2.25 and 3, indicated by bolder to more transparent dashed orange lines, respectively) are extrapolated to obtain the PEA estimate (solid orange lines). Using OBP with 5 Trotter steps backpropagated, the polarization can be measured to a higher accuracy in deep circuits (blue lines). The insets highlight the qubits of ibm_kyiv used to represent the spins. The qubits are initialized in either $\ket{0}$ (green circles) or $\ket{1}$ (red circles). b/d) Dynamics of several individual $Z_i$ under the XY model. The vertical dashed lines indicate Trotter steps at which the expectation values are measured. The orange scatter points indicate results from measurement at 5, 15, and 20 Trotter steps and applying PEA without OBP. The OBP framework helps recover the dynamics of intermediate time values (blue scatter points) from these coarse measurement data. The results agree with the reference values (solid gray lines) obtained via an MPS simulation. All error bars shown were obtained through bootstrapping with 100 batches, and are shown at a 2-$\sigma$ confidence. The shaded blue region represents the additional $L_2$ error bound due to the classical approximation of the backpropagated observable.
• Figure 5: