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arXiv:2406.02501 · Physical Review X

Computational Power of Random Quantum Circuits in Arbitrary Geometries

Matthew P. DeCross, Reza Haghshenas, Mingzhe Liu, Enrico Rinaldi, Johnnie Gray, Yuri Alexeev, Charles H. Baldwin, John P. Bartolotta, Matthew J. Bohn, Eli Chertkov, J. Cline, J. Colina, Davide DelVento, Joan Dreiling, C. Foltz, John Gaebler, Thomas M. Gatterman, C. N. Gilbreth, J. Giles, Dan Gresh, A. Hall, Aaron M. Hankin, A. Hansen, Nathan Hewitt, Ian M. Hoffman, C. A. Holliman, Ross B. Hutson, T. Jacobs, J. Johansen, P. J. Lee, Erik B Lehman, Dominic Lucchetti, Danylo Lykov, Ivaylo S. Madjarov, Brian Mathewson, Karl Mayer, M. Mills, Pradeep Niroula, J. M. Pino, Conrad Roman, Michael Schecter, P. E. Siegfried, Bruce G. Tiemann, Curtis Volin, J. Walker, Ruslan Shaydulin, Marco Pistoia, Steven A. Moses, David Hayes, Brian Neyenhuis, Russell Stutz, Michael Foss‐Feig

Published May 13, 2025• 45 citations
arXivHTMLPDF

Abstract

Empirical evidence for a gap between the computational powers of classical and quantum computers has been provided by experiments that sample the output distributions of two-dimensional quantum circuits. Many attempts to close this gap have utilized classical simulations based on tensor network techniques, and their limitations shed light on the improvements to quantum hardware required to frustrate classical simulability. In particular, quantum computers having in excess of approximately 50 qubits are primarily vulnerable to classical simulation due to restrictions on their gate fidelity and their connectivity, the latter determining how many gates are required (and, therefore, how much infidelity is suffered) in generating highly entangled states. Here, we describe recent hardware upgrades to Quantinuum’s H2 quantum computer, enabling it to operate on up to 56 qubits with arbitrary connectivity and 99.843(5)% two-qubit gate fidelity. We define a class of circuits with random geometries that become hard to classically simulate in very low depth and implement them utilizing the flexible connectivity of H2. A careful analysis demonstrating the fast saturation of classical simulation complexity with depth indicates that H2 can yield data well beyond the reach of state-of-the art classical simulation methods at unprecedented fidelities. We find that the considerable difficulty of classically simulating H2 is likely limited only by qubit number, demonstrating the promise and scalability of the quantum charge-coupled device architecture as continued progress is made toward building larger machines.

DOI
10.1103/physrevx.15.021052

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•Feb 9, 2026•0 citations
Empirical evidence for a gap between the computational powers of classical and quantum computers has been provided by experiments that sample the output distributions of two-dimensional quantum circuits. Many attempts to close this gap have utilized classical simulations based on tensor network techniques, and their limitations shed light on the improvements to quantum hardware required to frustrate classical simulability. In particular, quantum computers having in excess of approximately 50 qubits are primarily vulnerable to classical simulation due to restrictions on their gate fidelity and their connectivity, the latter determining how many gates are required (and, therefore, how much infidelity is suffered) in generating highly entangled states. Here, we describe recent hardware upgrades to Quantinuum’s H2 quantum computer, enabling it to operate on up to 56 qubits with arbitrary connectivity and 99.843(5)% two-qubit gate fidelity. We define a class of circuits with random geometries that become hard to classically simulate in very low depth and implement them utilizing the flexible connectivity of H2. A careful analysis demonstrating the fast saturation of classical simulation complexity with depth indicates that H2 can yield data well beyond the reach of state-of-the art classical simulation methods at unprecedented fidelities. We find that the considerable difficulty of classically simulating H2 is likely limited only by qubit number, demonstrating the promise and scalability of the quantum charge-coupled device architecture as continued progress is made toward building larger machines.
An ultimate goal of quantum computing is to perform calculations beyond the reach of any classical computer.It is therefore imperative that useful quantum computers be very difficult to simulate classically, otherwise classical computers could be used for the applications envisioned for the quantum ones.Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits N or the depth D of the circuit.This difficulty has triggered recent experiments on deep, random circuits that aim to demonstrate that quantum devices may already perform tasks beyond the reach of classical computing.These real quantum computing devices, however, suffer from many sources of decoherence and imprecision which limit the degree of entanglement that can actually be reached to a fraction of its theoretical maximum.They are characterized by an exponentially decaying fidelity F ∼ ð1 -ϵÞ ND with an error rate ϵ per operation as small as ≈1% for current devices with several dozen qubits or even smaller for smaller devices.In this work, we provide new insight on the computing capabilities of real quantum computers by demonstrating that they can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer.Our algorithms compress the representations of quantum wave functions using matrix product states, which are able to capture states with low to moderate entanglement very accurately.This compression introduces a finite error rate ϵ so that the algorithms closely mimic the behavior of real quantum computing devices.The computing time of our algorithm increases only linearly with N and D in sharp contrast with exact simulation algorithms.We illustrate our algorithms with simulations of random circuits for qubits connected in both one-and twodimensional lattices.We find that ϵ can be decreased at a polynomial cost in computing power down to a minimum error ϵ ∞ .Getting below ϵ ∞ requires computing resources that increase exponentially with ϵ ∞ =ϵ.For a two-dimensional array of N ¼ 54 qubits and a circuit with control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours.For more complex gates such as a SWAP gate followed by a controlled rotation, the error rate increases by a factor 3 for similar computing time.Our results suggest that, despite the high fidelity reached by quantum devices, only a tiny fraction ð∼10 -8 Þ of the system Hilbert space is actually being exploited.
Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond the capabilities of known classical algorithms on state-of-the-art computers, achieving the first milestone of so-called quantum supremacy. This has stimulated recent progress in classical algorithms to simulate quantum circuits. Classical simulations are also necessary to approximate the fidelity of multiqubit quantum computers using cross entropy benchmarking. Here we present numerical results of a classical simulation algorithm to sample universal random circuits, on a single workstation, with more qubits and depth than previously reported. For example, circuits with 5×95 \times 95×9 qubits of depth 37, 7×87 \times 87×8 qubits of depth 27, and 10×(κ>10)10 \times (κ> 10)10×(κ>10)) qubits of depth 19 are all easy to sample. We also show up to what depth the sampling, or estimation of observables, is trivially parallelizable. The algorithm is related to the "Feynmann path" method to simulate quantum circuits. For low-depth circuits, the algorithm scales exponentially in the depth times the smaller lateral dimension, or the treewidth, as explained in Boixo et. al., and therefore confirms the bounds in that paper. In particular, circuits with 7×77 \times 77×7 qubits and depth 40 remain currently out of reach. Follow up work on a supercomputer environment will tighten this bound.
We report on Quantinuum Helios, a 98-qubit trapped-ion quantum processor based on the quantum charge-coupled device (QCCD) architecture. Helios features 137^{137}137Ba+^{+}+ hyperfine qubits, all-to-all connectivity enabled by a rotatable ion storage ring connecting two quantum operation regions by a junction, speed improvements from parallelized operations, and a new software stack with real-time compilation of dynamic programs. Averaged over all operational zones in the system, we achieve average infidelities of 2.5(1)×10−52.5(1)\times10^{-5}2.5(1)×10−5 for single-qubit gates, 7.9(2)×10−47.9(2)\times10^{-4}7.9(2)×10−4 for two-qubit gates, and 4.8(6)×10−44.8(6)\times10^{-4}4.8(6)×10−4 for state preparation and measurement, none of which are fundamentally limited and likely able to be improved. These component infidelities are predictive of system-level performance in both random Clifford circuits and random circuit sampling, the latter demonstrating that Helios operates well beyond the reach of classical simulation and establishes a new frontier of fidelity and complexity for quantum computers.
As Moore's law reaches its limits, quantum computers are emerging with the promise of dramatically outperforming classical computers. We have witnessed the advent of quantum processors with over 505050 quantum bits (qubits), which are expected to be beyond the reach of classical simulation. Quantum supremacy is the event at which the old Extended Church-Turing Thesis is overturned: A quantum computer performs a task that is practically impossible for any classical (super)computer. The demonstration requires both a solid theoretical guarantee and an experimental realization. The lead candidate is Random Circuit Sampling (RCS), which is the task of sampling from the output distribution of random quantum circuits. Google recently announced a 53−53-53−qubit experimental demonstration of RCS. Soon after, classical algorithms appeared that challenge the supremacy of random circuits by estimating their outputs. How hard is it to classically simulate the output of random quantum circuits? We prove that estimating the output probabilities of random quantum circuits is formidably hard (#P\#P#P-Hard) for any classical computer. This makes RCS the strongest candidate for demonstrating quantum supremacy relative to all other proposals. The robustness to the estimation error that we prove may serve as a new hardness criterion for the performance of classical algorithms. To achieve this, we introduce the Cayley path interpolation between any two gates of a quantum computation and convolve recent advances in quantum complexity and information with probability and random matrices. Furthermore, we apply algebraic geometry to generalize the well-known Berlekamp-Welch algorithm that is widely used in coding theory and cryptography. Our results imply that there is an exponential hardness barrier for the classical simulation of most quantum circuits.
As quantum computers improve in the number of qubits and fidelity, the question of when they surpass state-of-the-art classical computation for a well-defined computational task is attracting much attention. The leading candidate task for this milestone entails sampling from the output distribution defined by a random quantum circuit. We develop a massively-parallel simulation tool Rollright that does not require inter-process communication (IPC) or proprietary hardware. We also develop two ways to trade circuit fidelity for computational speedups, so as to match the fidelity of a given quantum computer --- a task previously thought impossible. We report massive speedups for the sampling task over prior software from Microsoft, IBM, Alibaba and Google, as well as supercomputer and GPU-based simulations. By using publicly available Google Cloud Computing, we price such simulations and enable comparisons by total cost across hardware platforms. We simulate approximate sampling from the output of a circuit with 7x8 qubits and depth 1+40+1 by producing one million bitstring probabilities with fidelity 0.5%, at an estimated cost of $35184. The simulation costs scale linearly with fidelity, and using this scaling we estimate that extending circuit depth to 1+48+1 increases costs to one million dollars. Scaling the simulation to 10M bitstring probabilities needed for sampling 1M bitstrings helps comparing simulation to quantum computers. We describe refinements in benchmarks that slow down leading simulators, halving the circuit depth that can be simulated within the same time.
As quantum devices grow in size while remaining affected by noise, a key challenge is to determine when and how they can surpass classical computation on practical tasks. This requires the development of highly efficient classical simulation algorithms. Pauli backpropagation algorithms are among the most promising, having shown efficiency in simulating certain classes of parameterized quantum circuits, strong candidates for near‑term quantum advantage, under random circuit assumptions and depolarizing noise. However, their efficiency had not been demonstrated for more realistic nonunital noise models, such as amplitude damping, which better capture the behaviour of current hardware. We close this gap by adapting Pauli backpropagation to most single‑qubit noise channels, including nonunital noise, and proving that it remains efficient under these conditions. Our algorithm is parallelizable, memory‑efficient, and requires no assumptions on the locality of the observable, the geometry of the hardware, or the depth of the circuit. While hardware noise can strongly affect computational power in highly structured, fault‑tolerant circuits, our findings indicate little difference for most near‑term variational architectures. This establishes Pauli backpropagation as an efficient method for simulating noisy quantum devices under realistic noise assumptions.While these advances improve our ability to reproduce noisy expectation values of observables, many practical applications, including combinatorial optimization, require direct access to the bitstrings output by quantum circuits, as these encode candidate solutions. Classical replication of such noisy samples has so far been limited beyond the random circuit setting. To bridge this gap, we first focus on problems whose objective functions depend only on two‑body correlations, such as Max‑Cut, and apply Gaussian randomized rounding to the circuit's two‑qubit marginals. This produces classical samples whose expected cost is provably close to that of the noisy quantum device. Large‑scale simulations and experiments on IBMQ hardware confirm that the rounded samples reproduce the full energy distribution, with similar behaviour under various noise models. Coupled with Pauli backpropagation for efficiently estimating noisy marginals, this yields an end‑to‑end classical sampler operating in polynomial time for a broad class of variational circuits. Our results therefore show that sufficiently noisy quantum circuits for combinatorial optimization can be sampled from classically, clarifying where near‑term hardware may or may not outperform classical algorithms.Finally , we go beyond one and two‑body marginals and present an algorithm that samples valid bitstrings directly from a maximum‑entropy distribution matching the given expectation values of the quantum circuit. This method does not rely on the locality or sparsity of the problem's Hamiltonian and performs well across a variety of quantum circuits, including deep circuits on many qubits. When combined with Pauli propagation or error mitigation, it provides a practical pipeline for obtaining samples from quantum circuits in the near term, where full error correction might not yet be available.