Abstract We present an empirical analysis of the scaling of the minimal quantum circuit depth required for a variational quantum simulation (VQS) method to obtain a solution to the time evolution of a quantum system within a predefined error tolerance. In a comparison against a non-variational method based on Trotterized time evolution, we observe similar scaling behaviours of the depth requirements of VQS and Trotter when the size of the system and the simulated time grow linearly. Results are also put into perspective by discussing the corresponding classical complexity required for VQS. Our results allow us to identify a possible advantage region for VQS over Trotterization.
Abstract We present an empirical analysis of the scaling of the minimal quantum circuit depth required for a variational quantum simulation (VQS) method to obtain a solution to the time evolution of a quantum system within a predefined error tolerance. In a comparison against a non-variational method based on Trotterized time evolution, we observe similar scaling behaviours of the depth requirements of VQS and Trotter when the size of the system and the simulated time grow linearly. Results are also put into perspective by discussing the corresponding classical complexity required for VQS. Our results allow us to identify a possible advantage region for VQS over Trotterization.
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.
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.
Simulating time evolution of quantum systems is one of the most promising applications of quantum computing and also appears as a subroutine in many applications such as Green's function methods. In the current era of NISQ machines we assess the state of algorithms for simulating time dynamics with limited resources. We propose the Jaynes-Cummings model and extensions to it as useful toy models to investigate time evolution algorithms on near-term quantum computers. Using these simple models, direct Trotterisation of the time evolution operator produces deep circuits, requiring coherence times out of reach on current NISQ hardware. Therefore we test two alternative responses to this problem: variational compilation of the time evolution operator, and variational quantum simulation of the wavefunction ansatz. We demonstrate numerically to what extent these methods are successful in time evolving this system. The costs in terms of circuit depth and number of measurements are compared quantitatively, along with other drawbacks and advantages of each method. We find that computational requirements for both methods make them suitable for performing time evolution simulations of our models on NISQ hardware. Our results also indicate that variational quantum compilation produces more accurate results than variational quantum simulation, at the cost of a larger number of measurements.
Simulating quantum circuits using classical computers can accelerate the development and validation of quantum algorithms. Our newly developed algorithm, variational quantum search (VQS), has shown an exponential advantage over Grover's algorithm in the range from 5 to 26 qubits, in terms of circuit depth, for searching unstructured databases. We need to further validate the VQS for more than 26 qubits. Numerous simulators have been developed. However, it is not clear which simulator is most suitable for executing VQS with many qubits. To solve this issue, we implement a typical quantum circuit used in VQS on eight mainstream simulators. Results show that the time and memory required by most simulators increase exponentially with the number of qubits and that Pennylane with GPU and Qulacs are the most suitable simulators for executing VQS efficiently. Our results aid researchers in selecting suitable quantum simulators without the need for exhaustive implementation, and we have made our codes available for community contributions.
Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter approximation of the time-evolution operator. This approach typically relies on deep circuits and is therefore hampered by the substantial limitations of available noisy and near-term quantum hardware. On the other hand, variational quantum algorithms (VQAs) have become an indispensable alternative, enabling small-scale simulations on present-day hardware. However, despite the recent development of VQAs for quantum dynamics, a detailed assessment of their efficiency and scalability is yet to be presented. To fill this gap, we applied a VQA based on McLachlan's principle to simulate the dynamics of a spin-boson model subject to varying levels of realistic hardware noise as well as in different physical regimes, and discuss the algorithm's accuracy and scaling behavior as a function of system size. We observe a good performance of the variational approach used in combination with a general, physically motivated wave function ansatz, and compare it to the conventional first-order Trotter evolution. Finally, based on this, we make scaling predictions for the simulation of a classically intractable system. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage for the solution of time-dependent problems.
We propose a variational alternative to the Trotter-Suzuki decomposition that provides greater control over errors while preserving the unitary structure of time evolution. The variational parameters in our ansatz are derived from a global action principle, where Euler-Lagrange equations govern their optimal dynamics. Unlike conventional wavefunction-based variational methods, our approach specifically targets the time evolution operation and this allows a single set of optimized parameters to be applied to any initial state for a fixed Hamiltonian avoiding costly optimization procedures. Our method outperforms the standard Trotter-Suzuki formulas, typically achieving higher accuracy than higher-order Suzuki schemes. This translates directly to quantum computing applications, where it enables the design of quantum circuits with fewer gates which reduces noise and improves precision. Although we focus on quantum dynamics, the method is broadly applicable to problems involving general time-evolution operators. Applied to various model Hamiltonians, our approach reduces errors by factors of 2 to 5 compared to Trotter-Suzuki decompositions, demonstrating its promise for accurate quantum simulation with improved efficiency. In certain cases, the variational ansatz achieves higher accuracy than more complex higher-order Suzuki formulas while reducing the gate count by nearly half within a single circuit layer. Furthermore, we derive approximate analytical expressions for the variational parameters up to cubic order in time, valid for generic Hamiltonians. These approximations enable long-time quantum simulations with improved accuracy over equivalent Suzuki decompositions, providing ready-to-use evolution formulas that match Suzuki's gate complexity while delivering better performance.