Improved quantum computation using operator backpropagation
Bryce Fuller, Minh C. Tran, Danylo Lykov, Caleb Johnson, Max Rossmannek, Ken Xuan Wei, Andre He, Young‐Seok Kim, DinhDuy Vu, Kunal Sharma, Yuri Alexeev, Abhinav Kandala, Antonio Mezzacapo
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 and enable the recovery of expectation values at intermediate times throughout the classically backpropagated circuit, 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.
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 and enable the recovery of expectation values at intermediate times throughout the classically backpropagated circuit, 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.
Recent breakthroughs have opened the possibility of intermediate-scale quantum computing with tens to hundreds of qubits, and shown the potential for solving classical challenging problems, such as in chemistry and condensed matter physics. However, the high accuracy needed to surpass classical computers poses a critical demand on the circuit depth, which is severely limited by the non-negligible gate infidelity, currently around 0.1%-1%. The limited circuit depth places restrictions on the performance of variational quantum algorithms (VQA) and prevents VQAs from exploring desired nontrivial quantum states. To resolve this problem, we propose a paradigm of Schrödinger-Heisenberg variational quantum algorithms (SHVQA). Using SHVQA, the expectation values of operators on states that require very deep circuits to prepare can now be efficiently measured by rather shallow circuits. The idea is to incorporate a virtual Heisenberg circuit, which acts effectively on the measurement observables, into a real shallow Schrödinger circuit, which is implemented realistically on the quantum hardware. We choose a Clifford virtual circuit, whose effect on the Hamiltonian can be seen as efficient classical processing. Yet, it greatly enlarges the state's expressivity, realizing much larger unitary t designs. Our method enables accurate quantum simulation and computation that otherwise are only achievable with much deeper circuits or more accurate operations conventionally. This has been verified in our numerical experiments for a better approximation of random states, higher-fidelity solutions to the XXZ model, and the electronic structure Hamiltonians of small molecules. Thus, together with effective quantum error mitigation, our work paves the way for realizing accurate quantum computing algorithms with near-term quantum devices.
Computational methods are central to modern physics, yet many problems, especially those involving Quantum Many-Body Systems (QMBSs), remain challenging at scale. The core difficulty of simulating QMBSs is the exponential growth of the solution space with system size. This renders exact treatments intractable and pushes approximate methods to their limits when tackling two- and higher-dimensional systems and long-time dynamics. Quantum computers offer a quantum-native platform for representing and manipulating such systems. Recent algorithmic and technological advancements have enabled experiments on quantum processors in the quantum utility regime, a scale at which exact classical reproduction is no longer feasible. This represents the first step towards realizing quantum advantage, where quantum computers solve specific, practically relevant problems more efficiently than any classical approach. While advancing towards the goal of quantum advantage, useful computations on quantum processors are expected to be realised through hybrid quantum-classical algorithms, which combine quantum experiments and classical computational resources. A prominent example are variational quantum algorithms, where a Parametrized Quantum Circuit (PQC) prepares a trial quantum state whose parameters are iteratively updated by a classical optimisation routine to minimize a target cost function (e.g., the energy of a target Hamiltonian). However, the optimisation of such PQCs is affected by trainability issues, such as barren plateaus in the parameter landscape, which can render the optimisation increasingly difficult for larger system sizes. This motivates the development of novel hybrid approaches overcoming the limits of PQCs. In this thesis, we introduce a new class of approaches based on hybrid Tensor Networks (TNs). A hybrid TN is obtained by integrating quantum states prepared on a quantum processor within TNs -- one of the state-of-the-art approaches for simulating quantum systems. We first show how hybrid TNs can be used for ground-state searches within a variational setting. Then, we further extend the framework to non-variational hybrid TN ansätze for which the quantum tensors are prepared via quantum annealing and subsequently post-processed by classical TNs. In the context of quantum machine learning, which also relies on PQCs, we further investigate heuristic best practices that mitigate the same exponential concentration effects in kernel-based models for anomaly detection. Another class of hybrid algorithms are quantum error-mitigation workflows, where classical post-processing is used to reconstruct expectation values from noise-affected circuit executions. Such methods generally feature a trade-off between the bias and the variance: the price to pay for mitigating noise in a bias-free fashion is that the number of measurements usually grows exponentially with the depth of the circuit in order to achieve a fixed accuracy. In this thesis, we develop a low-resource error-mitigation strategy that reduces the measurement overhead in practical digital Hamiltonian-simulation workflows. We apply this method to Trotterized dynamics in a lattice-gauge-theory model, enabling the extraction of expectation values with improved accuracy at time-scales that allow the observation of post-collision scattering dynamics.
Simulating dynamics of physical systems is a key application of quantum computing, with potential impact in fields such as condensed matter physics and quantum chemistry. However, current quantum algorithms for Hamiltonian simulation yield results that are inadequate for real use cases and suffer from lengthy execution times when implemented on near-term quantum hardware. In this work, we introduce a framework for evaluating the performance of quantum simulation algorithms, focusing on the computation of observables, such as energy expectation values. Our framework provides end-to-end demonstrations of algorithmic optimizations that utilize Pauli term groups based onk-commutativity, generate customized Clifford measurement circuits, and implement weighted shot distribution strategies across these groups. These demonstrations span multiple quantum execution environments, allowing us to identify critical factors influencing runtime and solution accuracy. We integrate enhancements into the QED-C Application-Oriented Benchmark suite, utilizing problem instances from the opensource HamLib collection. Our results demonstrate a27.1%error reduction through Pauli grouping methods, with an additional 37.6% improvement from the optimized shot distribution strategy. Our framework provides an essential tool for advancing quantum simulation performance using algorithmic optimization techniques, enabling systematic evaluation of improvements that could maximize near-term quantum computers’ capabilities and advance practical quantum utility as hardware evolves.
Abstract Quantum algorithms on noisy intermediate‐scale quantum (NISQ) devices are expected to soon simulate quantum systems that are classically intractable. However, the non‐negligible gate error present on NISQ devices impedes the implementation of many purely quantum algorithms, necessitating the use of hybrid quantum‐classical algorithms. One such hybrid quantum‐classical algorithm, is based upon quantum computed Hamiltonian moments , with respect to quantum state . In this tutorial review, we will give a brief review of these quantum algorithms with focuses on the typical ways of computing Hamiltonian moments using quantum hardware and improving the accuracy of the estimated state energies based on the quantum computed moments. We also present a computation of the Hamiltonian moments of a four‐site Heisenberg model and compute the energy and magnetization of the model utilizing the imaginary time evolution on current IBM‐Q hardware. Finally, we discuss some possible developments and applications of Hamiltonian moment methods.
Decoherence imposes severe limits on quantum circuit depth, motivating methods to reduce circuit depth. Operator backpropagation has recently been proposed as a way to lower circuit depth by classically backpropagating observables through subcircuits. While it lowers the circuit depth, it also significantly increases the number of backpropagated observables that must be measured. This calls for an efficient measurement protocol tailored to these backpropagated observables. Moreover, errors arise both during the backpropagation procedure and during measurement, and they accumulate in the final computation. No prior work has examined how to integrate operator backpropagation with advanced measurement protocols, and how to jointly optimize the combined process.
In this paper, we introduce a comprehensive framework to determine the appropriate measurement protocol to achieve a desired level of accuracy with the fewest number of measurements based on the number of qubit-wise commuting (QWC) groups formed from operator backpropagation. We analyze and categorize the primary sources of error that arise through our workflow, including error incurred from the backpropagation algorithm, truncation error, and shot noise variation. We also examine truncation strategies to reduce the number of backpropagated observables by characterizing the structure of the set of backpropagated observables.
The study of quantum many-body systems is crucial to our understanding of nature at the microscopic scale. As analytical solutions quickly become intractable, the development of efficient computational simulation techniques is indispensable to predict the behaviour of quantum systems. This task is particularly challenging as the dimension of the underlying Hilbert space grows exponentially with the system size. While classical computational methods have achieved remarkable success in specific regimes, their applicability is fundamentally limited by this exponential scaling. Quantum computers offer a natural platform for representing and simulating quantum systems, but current devices are constrained by noise and limited qubit counts, drastically restricting the algorithms that can be run. In this context, hybrid quantum-classical algorithms provide a promising approach to combine the strengths of both paradigms. This thesis develops hybrid quantum-classical algorithms for the simulation of quantum many-body systems. The key idea is to perform classically difficult tasks on near-term quantum devices, while leveraging classical computational power to control, correct, and extend the obtained results. Within this framework, we introduce two complementary approaches for the simulation of quantum dynamics. The first method employs circuit knitting techniques to distribute variational quantum simulations across multiple quantum devices. The entanglement between weakly coupled clusters of highly correlated spins is recovered in classical post-processing, where the overhead is controlled through a constrained optimisation problem. The second method combines a simplified, Trotterised quantum dynamics simulation with a parametrised classical ansatz trained to correct for the approximations made on the quantum device. We show that this method can be employed to correct Trotter errors, implement long-range gates with hardware-efficient circuits, or simulate larger systems by incorporating additional degrees of freedom purely classically. In addition, we present a hybrid approach to computing thermal properties of quantum many-body systems via the Quantum Finite Temperature Lanczos Method. A Krylov subspace is generated on a quantum computer by combining the efficient preparation of typical states with real-time evolution. A classical computer then diagonalises the system in the low-dimensional subspace and estimates the finite-temperature expectation values through trace estimation. The results presented in this thesis demonstrate that hybrid quantum-classical algorithms can expand the range of systems that can be studied on near-term quantum devices. Conversely, we show that incorporating quantum subroutines into classical simulation techniques can improve their efficiency. These results suggest that, rather than treating classical and quantum simulation as competing paradigms, their careful integration can provide a more effective route toward practical quantum many-body simulation.