We describe and analyze algorithms for classically simulating measurement of an n-qubit quantum state in the standard basis, that is, sampling a bit string from the probability distribution determined by the Born rule. Our algorithms reduce the sampling task to computing poly(n) amplitudes of n-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. Two classes of quantum states are considered: output states of polynomial-size quantum circuits, and ground states of local Hamiltonians with an inverse polynomial spectral gap. We show that our algorithms can significantly accelerate quantum circuit simulations based on tensor network contraction or low-rank stabilizer decompositions. As another striking consequence we obtain the first efficient classical simulation algorithm for measurement-based quantum computation with the surface code resource state on any planar graph and any schedule of measurements.
We describe and analyze algorithms for classically simulating measurement of an n-qubit quantum state in the standard basis, that is, sampling a bit string from the probability distribution determined by the Born rule. Our algorithms reduce the sampling task to computing poly(n) amplitudes of n-qubit states; unlike previously known techniques they do not require computation of marginal probabilities. Two classes of quantum states are considered: output states of polynomial-size quantum circuits, and ground states of local Hamiltonians with an inverse polynomial spectral gap. We show that our algorithms can significantly accelerate quantum circuit simulations based on tensor network contraction or low-rank stabilizer decompositions. As another striking consequence we obtain the first efficient classical simulation algorithm for measurement-based quantum computation with the surface code resource state on any planar graph and any schedule of measurements.
We consider the efficiency of classically simulating measurement-based quantum computation on surface-code states. We devise a method for calculating the elements of the probability distribution for the classical output of the quantum computation. The operational cost of this method is polynomial in the size of the surface-code state, but in the worst case scales as 22g in the genus g of the surface embedding the code. However, there are states in the code space for which the simulation becomes efficient. In general, the simulation cost is exponential in the entanglement contained in a certain effective state, capturing the encoded state, the encoding, and the local postmeasurement states. The same efficiencies hold, with additional assumptions on the temporal order of measurements and on the tessellations of the code surfaces, for the harder task of sampling from the distribution of the computational output.
Noise in quantum operations often negates the advantage of quantum computation. However, most classical simulations of quantum computers calculate the ideal probability amplitudes by either storing full state vectors or using sophisticated tensor-network contractions. Here we investigate sampling-based classical simulation methods for noisy quantum circuits. Specifically, we characterize the simulation costs of two major schemes, stabilizer-state sampling of magic states and Heisenberg propagation, for quantum circuits subject to stochastic Pauli noise, such as depolarizing and dephasing noise. To this end, we introduce several techniques for the stabilizer-state sampling to reduce the simulation costs under such noise. It is revealed that in the low-noise regime, stabilizer-state sampling results in a smaller sampling cost, while Heisenberg propagation is better in the high-noise regime. Furthermore, for a high depolarizing noise rate of approximately 10%, these methods provide better scaling than that given by the low-rank stabilizer decomposition. We believe that this knowledge of classical simulation costs is useful to achieve possible quantum advantage on near-term noisy quantum devices as well as efficient classical simulation methods.
Quantum circuit simulators running on classical computers offer a vital platform for designing, testing, and optimizing quantum algorithms, driving innovation despite limited access to real quantum hardware. However, their scalability is inherently constrained by exponential memory and computational overhead, which restricts accurate simulation of large-scale quantum circuits and often results in approximate output distributions. Here, we propose an exact sampling algorithm that integrates tensor network contraction techniques with a Markov process, wherein a classical state evolves according to the local structure of the quantum circuit. As a demonstration, we target the challenge of generating samples from ideal and noisy QAOA circuits with up to 476 qubits, incorporating both depolarizing and amplitude damping noise models. These results enable further validation of several assumptions and conjectures at a scale previously out of reach, significantly expanding the scope of classical simulation in quantum algorithm research.
Various algorithms have been developed to simulate quantum circuits on classical hardware. Among the most prominent are approaches based on \emph{stabilizer decompositions} and \emph{tensor network contraction}. In this work, we present a unified framework that bridges these two approaches, placing them under a common formalism. Using this, we present two new algorithms to simulate an n-qubit circuit C: one that runs in O~(Ttw(C)) time and the other in O~(Tγ⋅tw(C)) time, where tw(C) and rw(C) refer to the the tree-width and rank-width, respectively, of a tensor network associated to C, T is the number of non-Clifford gates in C, and γ≈3.42. The proposed algorithms are simple, only require a linear amount of memory, are trivially parallelizable, and interact nicely with ZX-diagram simplification routines. Furthermore, we introduce the refined complexity measures \emph{focused tree-width} and \emph{focused rank-width}, which are always at least as efficient as their standard equivalent; these can be directly applied within our simulation algorithms, allowing for a more precise upper bound on the run time.
We present two classical algorithms for the simulation of universal quantum circuits on n qubits constructed from c instances of Clifford gates and t arbitrary-angle Z-rotation gates such as T gates.Our algorithms complement each other by performing best in different parameter regimes.The ESTIMATE algorithm produces an additive precision estimate of the Born-rule probability of a chosen measurement outcome with the only source of run-time inefficiency being a linear dependence on the stabilizer extent (with scaling approximately equal to 1.17 t for T gates).Our algorithm is state of the art for this task: as an example, in approximately 13 h (on a standard desktop computer), we estimate the Born-rule probability to within an additive error of 0.03, for a 50-qubit, 60 non-Clifford gate quantum circuit with more than 2000 Clifford gates.Our second algorithm, COMPUTE, calculates the probability of a chosen measurement outcome to machine precision with run time O 2 t-r t , where r is an efficiently computable, circuit-specific quantity.With high probability, r is very close to min {t, n -w} for random circuits with many Clifford gates, where w is the number of measured qubits.COMPUTE can be effective in surprisingly challenging parameter regimes, e.g., we can randomly sample Clifford+T circuits with n = 55, w = 5, c = 10 5 , and t = 80 T gates, and then compute the Born-rule probability with a run time consistently less than 10 min using a single core of a standard desktop computer.We provide a C+Python implementation of our algorithms and benchmark them using random circuits, the hidden-shift algorithm, and the quantum approximate optimization algorithm.
Abstract The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions—a so-called standard form—which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus, enabling a significant decrease in circuit depth for certain applications. We show that groups of mutually-commuting operators can be implemented using fully-parallel, i.e. non-adaptive, measurements within our protocol. In addition, we discuss how groups of mutually commuting observables can be simulatenously measured by adjusting the resource state, rather than performing a costly basis transformation prior to the measurement as it is done in the circuit model. Finally, we demonstrate the utility of our technique on two examples of high practical relevance—the Quantum Approximate Optimization Algorithm and the Variational Quantum Eigensolver (VQE) for the ground-state energy estimation of the water molecule. For the VQE, we find a reduction of the depth by a factor of 4 to 5 using measurement patterns vs. the standard circuit model. At the same time, since we incorporate the simultaneous measurements, our patterns allow us to save shots by a factor of at least 3.5 compared to measuring Pauli strings individually in the circuit model.