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arXiv:2212.08609 · Physical Review A

Classically simulating intermediate-scale instantaneous quantum polynomial circuits through a random graph approach

Julien Codsi, John van de Wetering

Published Jan 14, 2025• 4 citations
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Abstract

Quantum advantage is a demonstration of a computation by a quantum computer that cannot be performed by the best classical computer in a reasonable time. A well-studied approach to demonstrating this on near-term quantum computers is to use random circuit sampling. It has been suggested that a good candidate for demonstrating quantum advantage with random circuit sampling is to use instantaneous quantum polynomial (IQP) circuits. These are quantum circuits where the unitary it implements is diagonal. In this paper, we introduce improved techniques for exactly classically simulating random IQP circuits. We find a simple algorithm to calculate an amplitude of an $n$-qubit IQP circuit with dense random two-qubit interactions in time $O(\frac{{log}^{2}n}{n}{2}^{n})$, which for sparse circuits (where each qubit interacts with $O(logn)$ other qubits) runs in $o({2}^{n}/\mathrm{poly}(n))$ for any given polynomial. Using a more complicated stabilizer decomposition approach we improve the algorithm for dense circuits to $O(\frac{{(logn)}^{4\ensuremath{-}\ensuremath{\beta}}}{{n}^{2\ensuremath{-}\ensuremath{\beta}}}{2}^{n})$ where $\ensuremath{\beta}\ensuremath{\approx}0.396$. We further discuss how our techniques also lead to improved simulation times for IQP circuits on restricted architectures. We benchmarked our main algorithm and found that we can simulate up to 50-qubit circuits in a couple of minutes on a laptop, with 68-qubit sparse circuits taking a couple of hours. We estimate dense 70-qubit circuits are in range for large computing clusters.

DOI
10.1103/physreva.111.012422

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Quantum advantage is a demonstration of a computation by a quantum computer that cannot be performed by the best classical computer in a reasonable time. A well-studied approach to demonstrating this on near-term quantum computers is to use random circuit sampling. It has been suggested that a good candidate for demonstrating quantum advantage with random circuit sampling is to use instantaneous quantum polynomial (IQP) circuits. These are quantum circuits where the unitary it implements is diagonal. In this paper, we introduce improved techniques for exactly classically simulating random IQP circuits. We find a simple algorithm to calculate an amplitude of an nnn-qubit IQP circuit with dense random two-qubit interactions in time O(log2nn2n)O(\frac{{log}^{2}n}{n}{2}^{n})O(nlog2n​2n), which for sparse circuits (where each qubit interacts with O(logn)O(logn)O(logn) other qubits) runs in o(2n/poly(n))o({2}^{n}/\mathrm{poly}(n))o(2n/poly(n)) for any given polynomial. Using a more complicated stabilizer decomposition approach we improve the algorithm for dense circuits to O((logn)4\ensuremath−\ensuremathβn2\ensuremath−\ensuremathβ2n)O(\frac{{(logn)}^{4\ensuremath{-}\ensuremath{\beta}}}{{n}^{2\ensuremath{-}\ensuremath{\beta}}}{2}^{n})O(n2\ensuremath−\ensuremathβ(logn)4\ensuremath−\ensuremathβ​2n) where \ensuremathβ\ensuremath≈0.396\ensuremath{\beta}\ensuremath{\approx}0.396\ensuremathβ\ensuremath≈0.396. We further discuss how our techniques also lead to improved simulation times for IQP circuits on restricted architectures. We benchmarked our main algorithm and found that we can simulate up to 50-qubit circuits in a couple of minutes on a laptop, with 68-qubit sparse circuits taking a couple of hours. We estimate dense 70-qubit circuits are in range for large computing clusters.
Sampling from the output distributions of quantum computations comprising only commuting gates, known as instantaneous quantum polynomial (IQP) computations, is believed to be intractable for classical computers, and hence this task has become a leading candidate for testing the capabilities of quantum devices. Here we demonstrate that for an arbitrary IQP circuit undergoing dephasing or depolarizing noise, whose depth is greater than a critical O (1) threshold, the output distribution can be efficiently sampled by a classical computer. Unlike other simulation algorithms for quantum supremacy tasks, we do not require assumptions on the circuit’s architecture, on anti-concentration properties, nor do we require Ω(log (n )) circuit depth. We take advantage of the fact that IQP circuits have deep sections of diagonal gates, which allows the noise to build up predictably and induce a large-scale breakdown of entanglement within the circuit. Our results suggest that quantum supremacy experiments based on IQP circuits may be more susceptible to classical simulation than previously thought. Furthermore, we show that the critical depth threshold of our algorithm is tight, and below this threshold there are noisy IQP circuits which are hard to sample from. Thus we demonstrate that noisy IQP circuits exhibit a phase transition in the computational complexity of sampling, as circuit depth is increased.
Instantaneous Quantum Polynomial-time (IQP) circuits are a candidate for demonstrating near-term quantum advantage, as their sampling task is believed to be classically hard in the ideal theoretical setting under standard complexity-theoretic assumptions. In noisy implementations, however, this hardness can disappear once circuit depth exceeds a noise-dependent critical threshold. We show that qubit connectivity is a key parameter in this transition, since sparse architectures require additional routing to implement long-range interactions, thereby increasing compiled circuit depth. To make this explicit, we present a connectivity-aware analysis of compiled IQP circuits. For a fixed abstract IQP instance, different hardware connectivity graphs yield different compiled depths and thus different effective positions relative to the noisy-IQP simulatability boundary. We quantify this architecture-dependent shift using the compiled depth overhead and the corresponding simulatability margin. We combine analytic depth estimates for sparse geometries, including the two-dimensional grid, with native-gateset-aware compilation experiments across seven hardware-grounded experimental device models derived from publicly available topologies. To compare these device models under a unified empirical framework, we approximate the effective noise level primarily through reported two-qubit gate error rates. This lets us compare how much effective noise sparse and fully connected architectures can tolerate for the same position relative to the noisy-IQP simulatability boundary. Our results show that sparse connectivity requires a lower effective noise level to sustain the same margin relative to the noisy-IQP simulatability boundary, and they provide a quantitative framework for determining when compiled IQP experiments are likely to remain outside, or instead enter, the classically simulatable regime.
The class of commuting quantum circuits known as IQP (instantaneous quantum polynomial-time) has been shown to be hard to simulate classically, assuming certain complexity-theoretic conjectures. Here we study the power of IQP circuits in the presence of physically motivated constraints. First, we show that there is a family of sparse IQP circuits that can be implemented on a square lattice of n qubits in depth O(sqrt(n) log n), and which is likely hard to simulate classically. Next, we show that, if an arbitrarily small constant amount of noise is applied to each qubit at the end of any IQP circuit whose output probability distribution is sufficiently anticoncentrated, there is a polynomial-time classical algorithm that simulates sampling from the resulting distribution, up to constant accuracy in total variation distance. However, we show that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form. These results demonstrate the challenges faced by experiments designed to demonstrate quantum supremacy over classical computation, and how these challenges can be overcome.
Efficient classical simulation of noisy intermediate-scale quantum (NISQ) circuits has been a topic of intense study over the past few years. The majority of results on efficient simulation assume that the circuits undergo some variant of unital noise or involve sufficient randomness. However, there are limited results for circuits undergoing non-unital noise in the absence of randomness. In this work, we present a polynomial-time classical algorithm to sample from the output distributions of amplitude-damped instantaneous quantum polynomial (IQP) circuits. Our algorithm works for circuits generated by arbitrary lll-local diagonal gates with depth d=Ω(log⁡(n))d = Ω(\log(n))d=Ω(log(n)), undergoing constant amplitude-damping noise.
Generative modeling is one of the most promising applications of quantum machine learning, yet training and deploying Quantum Generative Models (QGMs) on near-term hardware remains effectively intractable due to prohibitive gradient estimation and implementation costs. We propose a resource-efficient approach based on shallow Instantaneous Quantum Polynomial-time (IQP) circuits that circumvents these bottlenecks by leveraging efficient classical training while retaining the guarantee of sampling hardness. To validate this approach, we formalize graph generation as a hierarchy of physical correlations, allowing us to map abstract data features, such as edge density and bipartiteness, directly to the quantum observables required to learn them. We validate our protocol through demonstrations both on real hardware (from 282828 to 153153153 qubits) and simulations (282828 qubits). Results show that while global structural features exhibit significant degradation beyond 919191 qubits, our models achieve high-precision reproduction of local correlations, even up to 153153153 qubits. These findings establish shallow IQP circuits as a robust, scalable candidate for generative tasks on current Noisy Intermediate-Scale Quantum (NISQ) devices.
A central question in the theory of quantum advantage is whether there are quantum advantage protocols with similar resource requirements as random circuit sampling that are also verifiable just from the classical outputs of the quantum computation. Here, we develop the idea of simulation secrets for verifiable advantage. A verifier can use a simulation secret to evaluate a cross-entropy test faster than it would take a classical adversary to pass the test. We instantiate this idea using IQP circuits described by cubic polynomials with planted independent spaces. These correspond to the largest independent set in the orbit of a polynomial under the general linear group and yield a low-rank stabilizer decomposition of the corresponding state. We conjecture that large independent spaces are invisible to a computationally bounded adversary, and therefore they cannot exploit them to pass the protocol. A second conjecture regards the fine-grained complexity of producing samples that pass the cross-entropy test for uniformly random polynomials. Under these conjectures, our scheme results in a polynomial gap between the verification time and the time a classical adversary would need to pass the protocol---both are exponential. It has a potential application to generating classically certifiable randomness, since the output distributions have high min-entropy. We estimate that the planted polynomial scheme is implementable using 100 logical qubits at logical error rates around 10−610^{-6}10−6.