Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. Quantum phase estimation uses deep circuits and is infeasible without fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit the power of shallow circuits. These algorithms rely on the assumption that variational circuits can produce the desired result. Here, we combine quantum Monte Carlo with quantum computing and propose a quasi-exact algorithm for imaginary-time simulation and ground-state computing. Unlike variational algorithms, our algorithm always approaches the exact solution when the Trotter circuit depth increases. Even when the circuit is shallow, our algorithm can yield an accurate ground-state energy. Compared with quantum phase estimation, the conventional quasi-exact algorithm, our algorithm can reduce the Trotter step number by thousands of times. We verify this resilience to Trotterisation errors in numerical simulation of up to 20 qubits and theoretical analysis. Our results demonstrate that non-variational and exact quantum simulation is promising even without a fully fault-tolerant quantum computer.
Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. Quantum phase estimation uses deep circuits and is infeasible without fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit the power of shallow circuits. These algorithms rely on the assumption that variational circuits can produce the desired result. Here, we combine quantum Monte Carlo with quantum computing and propose a quasi-exact algorithm for imaginary-time simulation and ground-state computing. Unlike variational algorithms, our algorithm always approaches the exact solution when the Trotter circuit depth increases. Even when the circuit is shallow, our algorithm can yield an accurate ground-state energy. Compared with quantum phase estimation, the conventional quasi-exact algorithm, our algorithm can reduce the Trotter step number by thousands of times. We verify this resilience to Trotterisation errors in numerical simulation of up to 20 qubits and theoretical analysis. Our results demonstrate that non-variational and exact quantum simulation is promising even without a fully fault-tolerant quantum computer.
Computing the ground-state properties of quantum many-body systems is a promising application of near-term quantum hardware with a potential impact in many fields. The conventional algorithm quantum phase estimation uses deep circuits and requires fault-tolerant technologies. Many quantum simulation algorithms developed recently work in an inexact and variational manner to exploit shallow circuits. In this work, we combine quantum Monte Carlo with quantum computing and propose an algorithm for simulating the imaginary-time evolution and solving the ground-state problem. By sampling the real-time evolution operator with a random evolution time according to a modified Cauchy-Lorentz distribution, we can compute the expected value of an observable in imaginary-time evolution. Our algorithm approaches the exact solution given a circuit depth increasing polylogarithmically with the desired accuracy. Compared with quantum phase estimation, the Trotter step number, i.e. the circuit depth, can be thousands of times smaller to achieve the same accuracy in the ground-state energy. We verify the resilience to Trotterisation errors caused by the finite circuit depth in the numerical simulation of various models. The results show that Monte Carlo quantum simulation is promising even without a fully fault-tolerant quantum computer.
Understanding the dynamics of quantum systems is crucial in many areas of physics, but simulating many-body systems presents significant challenges due to the large Hilbert space to navigate and the exponential growth of computational overhead. Quantum computers offer a promising platform to overcome these challenges, particularly for simulating the time evolution with Hamiltonians. Trotterization is a widely used approach among available algorithms in this regard, and well suited for near-term quantum devices. However, it introduces algorithmic Trotter errors due to the non-commutativity of Hamiltonian components. Several techniques such as multi-product formulas have been developed to mitigate Trotter errors, but often require deep quantum circuits, which can introduce additional physical errors. In this work, we propose a resource-efficient scheme to reduce the algorithmic Trotter error with relatively shallow circuit depth. We develop a profiling method by introducing an auxiliary parameter to estimate the error effects in expectation values, enabling significant error suppression with a fixed number of Trotter steps. Our approach offers an efficient way of quantum simulation on near-term quantum processors with shallow circuits.
Abstract Determining the ground and excited state properties of materials is considered one of the most promising applications of quantum computers. On near-term hardware, the limiting constraint on such simulations is the requisite circuit depths and qubit numbers, which currently lie well beyond near-term capabilities. Here we develop a quantum algorithm which reduces the estimated cost of material simulations. For example, we obtain a circuit depth improvement by up to 6 orders of magnitude for a Trotter layer of time-dynamics simulation in the transition-metal oxide SrVO3compared with the best previous quantum algorithms. We achieve this by introducing a collection of connected techniques, including highly localised and physically compact representations of materials Hamiltonians in the Wannier basis, a hybrid fermion-to-qubit mapping, and an efficient circuit compiler. Combined together, these methods leverage locality of materials Hamiltonians and result in a design that generates quantum circuits with depth independent of the system’s size. Although the requisite resources for the quantum simulation of materials are still beyond current hardware, our results show that realistic simulation of specific properties may be feasible without necessarily requiring fully scalable, fault-tolerant quantum computers, providing quantum algorithm design incorporates deeper understanding of the target materials and applications.
Dynamical simulation of quantum many-body systems is a central task in quantum chemistry and requires efficient wavefunction representations. Although tensor-network and neural-network quantum states have achieved considerable success in ground-state calculations, entanglement growth hinders their application to quantum dynamics. Quantum computing may offer a route to quantum advantage, but algorithms such as Trotterization generally require fault-tolerant quantum computers, whereas near-term hardware supports only limited circuit sizes. Variational quantum simulation (VQS) instead represents the evolving state with a single shallow, fixed-size parameterized quantum circuit (PQC). Here, we introduce nonorthogonal variational quantum simulation (NOVQS), which applies linear combinations of parameterized quantum states to real- and imaginary-time evolutions. We design a shallow, hardware-friendly ansatz tailored to second-quantized electronic-structure Hamiltonians, together with resource-efficient protocols for measuring the matrices and vectors in the parameter equations of motion. Error analysis and resource estimation are also provided. Numerical simulations of hydrogen chains and the nitrogen molecule demonstrate that a collection of shallow, or even single-layer, PQCs can match or outperform a much deeper PQC in VQS. In particular, we identify a trade-off between circuit number and depth. NOVQS therefore provides a flexible route to enhancing wavefunction expressivity under circuit-depth constraints, making it a promising framework for quantum dynamics simulations on near-term quantum processors.
Quantum simulation is a promising way toward practical quantum advantage, but noise in current quantum hardware poses a significant obstacle. We prove that not only the physical error but also the algorithmic error in a single Trotter step decreases exponentially with the circuit depth. This theoretical finding is validated by our numerical results over various Hamiltonians, initial states, and noise channels. Furthermore, we derive the optimal number of Trotter steps and the noise requirement to guarantee total simulation precision. To explicitly show the requirements for robust quantum simulation, we plot a phase diagram of the accumulated error in terms of circuit depth and noise rate. At last, we demonstrate that our improved error analysis leads to significant resourcesaving for fault-tolerant Trotter circuits. By addressing these aspects, this work provides fresh and systematic insight on the practical quantum advantage through quantum simulation.
Implementing time-evolution operators in shallow quantum circuits is important for quantum simulations. The standard method of Trotterization requires a large number of gates to achieve practical accuracy. Variational Quantum Simulation (VQS) is an algorithm that calculates the time evolution of a quantum state and can be executed with shallower circuits than Trotterization. However, the operator obtained by VQS evolves only a fixed initial state and is not the time evolution operator itself. In this paper, we propose Variational Quantum Operator Simulation (VQOS), a method to realize time evolution operators in shallow quantum circuits. This method is based on the variational principle for operators and does not require the implementation of the desired Trotter decomposition of the time evolution operator. We performed numerical simulations of the VQOS algorithm and successfully implemented the time evolution operator for closed systems in a quantum circuit that is up to 5 times shallower than the Trotterization. By providing a more practical way to implement time evolution operators, VQOS increases the applicability of near-term quantum computers.