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Product formula approximations of the time-evolution operator on quantum computers are of great interest due to their simplicity, and good scaling with system size by exploiting commutativity between Hamiltonian terms. However, product formulas exhibit poor scaling with the time $t$ and error $epsilon$ of simulation as the gate cost of a single step scales exponentially with the order $m$ of accuracy. We introduce well-conditioned multiproduct formulas, which are a linear combination of product formulas, where a single step has polynomial cost $mathcal{O}(m^2log{(m)})$ and succeeds with probability $Omega(1/operatorname{log}^2{(m)})$. Our multiproduct formulas imply a simple and generic simulation algorithm that simultaneously exploits commutativity in arbitrary systems and has a worst-case cost $mathcal{O}(tlog^{2}{(t/epsilon)})$ which is optimal up to poly-logarithmic factors. In contrast, prior Trotter and post-Trotter Hamiltonian simulation algorithms realize only one of these two desirable features. A key technical result of independent interest is our solution to a conditioning problem in previous multiproduct formulas that amplified numerical errors by $e^{Omega(m)}$ in the classical setting, and led to a vanishing success probability $e^{-Omega(m)}$ in the quantum setting.
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Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in Hamiltonian simulation. We find that, contrary to our intuition, anti-commutative relations could also reduce the hardness of Hamiltonian simulation. Specifically, Hamiltonians with mutually anti-commutative terms are easy to simulate, as what happens with ones consisting of mutually commutative terms. Such a property is further utilized to reduce the algorithmic error or the gate complexity in the truncated Taylor series quantum algorithm for general problems. Moreover, we propose two modified linear combinations of unitaries methods tailored for Hamiltonians with different degrees of anti-commutation. We numerically verify that the proposed methods exploiting anti-commutative relations could significantly improve the simulation accuracy of electronic Hamiltonians. Our work sheds light on the roles of commutative and anti-commutative relations in simulating quantum systems.
Hamiltonian simulation is one of the most important problems in quantum computation, and quantum singular value transformation (QSVT) is an efficient way to simulate a general class of Hamiltonians. However, the QSVT circuit typically involves multiple ancilla qubits and multi-qubit control gates. We propose a drastically simplified quantum circuit called the minimal QSVT circuit, which uses only one ancilla qubit to simulate a class of $n$-qubit random Hamiltonians. We formulate a simple metric called the quantum unitary evolution score (QUES), which is a scalable quantum benchmark and can be verified without any need for classical computation. We demonstrate that QUES is directly related to the circuit fidelity, and the classical hardness of an associated quantum circuit sampling problem. Theoretical analysis suggests under suitable assumptions, there exists an optimal simulation time $t^{text{opt}}approx 4.81$, at which even a noisy quantum device may be sufficient to demonstrate the classical hardness.
We study how parallelism can speed up quantum simulation. A parallel quantum algorithm is proposed for simulating the dynamics of a large class of Hamiltonians with good sparse structures, called uniform-structured Hamiltonians, including various Hamiltonians of practical interest like local Hamiltonians and Pauli sums. Given the oracle access to the target sparse Hamiltonian, in both query and gate complexity, the running time of our parallel quantum simulation algorithm measured by the quantum circuit depth has a doubly (poly-)logarithmic dependence $operatorname{polylog}log(1/epsilon)$ on the simulation precision $epsilon$. This presents an exponential improvement over the dependence $operatorname{polylog}(1/epsilon)$ of previous optimal sparse Hamiltonian simulation algorithm without parallelism. To obtain this result, we introduce a novel notion of parallel quantum walk, based on Childs quantum walk. The target evolution unitary is approximated by a truncated Taylor series, which is obtained by combining these quantum walks in a parallel way. A lower bound $Omega(log log (1/epsilon))$ is established, showing that the $epsilon$-dependence of the gate depth achieved in this work cannot be significantly improved. Our algorithm is applied to simulating three physical models: the Heisenberg model, the Sachdev-Ye-Kitaev model and a quantum chemistry model in second quantization. By explicitly calculating the gate complexity for implementing the oracles, we show that on all these models, the total gate depth of our algorithm has a $operatorname{polylog}log(1/epsilon)$ dependence in the parallel setting.
The evaluation of the performance of adiabatic annealers is hindered by lack of efficient algorithms for simulating their behaviour. We exploit the analyticity of the standard model for the adiabatic quantum process to develop an efficient recursive method for its numerical simulation in case of both unitary and non-unitary evolution. Numerical simulations show distinctly different distributions for the most important figure of merit of adiabatic quantum computing --- the success probability --- in these two cases.
We use discrete-event simulation on a digital computer to study two different models of experimentally realizable quantum walks. The simulation models comply with Einstein locality, are as realistic as the one of the simple random walk in that the particles follow well-defined trajectories, are void of concepts such as particle-wave duality and wave-function collapse, and reproduce the quantum-theoretical results by means of a cause-and-effect, event-by-event process. Our simulation model for the quantum walk experiment presented in [C. Robens et al., Phys. Rev. X 5, 011003 (2015)] reproduces the result of that experiment. Therefore, the claim that the result of the experiment rigorously excludes (i.e., falsifies) any explanation of quantum transport based on classical, well-defined trajectories needs to be revised.
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