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arXiv:2205.09900 · npj Quantum Information

Estimating the randomness of quantum circuit ensembles up to 50 qubits

Minzhao Liu, Junyu Liu, Yuri Alexeev, Liang Jiang

Published Nov 24, 2022• 11 citations
arXivHTMLPDF

Abstract

Abstract Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any random unitaries has consequences on their complexity, expressibility, and trainability. To study this property of random circuits, we develop numerical protocols for estimating the frame potential, the distance between a given ensemble and the exact randomness. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high-performing using CPU and GPU parallelism. We study 1. local and parallel random circuits to verify the linear growth in complexity as stated by the Brown–Susskind conjecture, and; 2. hardware-efficient ansätze to shed light on its expressibility and the barren plateau problem in the context of variational algorithms. Our work shows that large-scale tensor network simulations could provide important hints toward open problems in quantum information science.

DOI
10.1038/s41534-022-00648-7

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Abstract Random quantum circuits have been utilized in the contexts of quantum supremacy demonstrations, variational quantum algorithms for chemistry and machine learning, and blackhole information. The ability of random circuits to approximate any random unitaries has consequences on their complexity, expressibility, and trainability. To study this property of random circuits, we develop numerical protocols for estimating the frame potential, the distance between a given ensemble and the exact randomness. Our tensor-network-based algorithm has polynomial complexity for shallow circuits and is high-performing using CPU and GPU parallelism. We study 1. local and parallel random circuits to verify the linear growth in complexity as stated by the Brown–Susskind conjecture, and; 2. hardware-efficient ansätze to shed light on its expressibility and the barren plateau problem in the context of variational algorithms. Our work shows that large-scale tensor network simulations could provide important hints toward open problems in quantum information science.
The ability to efficiently simulate random quantum circuits using a classical computer is increasingly important for developing noisy intermediate-scale quantum devices. Here, we present a tensor network states based algorithm specifically designed to compute amplitudes for random quantum circuits with arbitrary geometry. Singular value decomposition based compression together with a two-sided circuit evolution algorithm are used to further compress the resulting tensor network. To further accelerate the simulation, we also propose a heuristic algorithm to compute the optimal tensor contraction path. We demonstrate that our algorithm is up to 2 orders of magnitudes faster than the Schrödinger-Feynman algorithm for verifying random quantum circuits on the 53-qubit Sycamore processor, with circuit depths below 12. We also simulate larger random quantum circuits with up to 104 qubits, showing that this algorithm is an ideal tool to verify relatively shallow quantum circuits on near-term quantum computers.
Variational quantum algorithms (VQAs) are among the leading approaches for near-term quantum computing, yet their performance can degrade in barren plateau regimes characterized by vanishing gradients. A widely held intuition is that increasing circuit expressivity, often associated with random-state behavior, leads to a loss of trainability. Existing results show that sufficiently random circuits can lead to barren plateaus. Here we show that standard statistical signatures of randomness can emerge well before this regime, without degrading trainability. We demonstrate this behavior in structured variational circuits applied to the one-dimensional cluster-Ising model and a generalized toric code Hamiltonian. To characterize state complexity, we analyze Porter-Thomas statistics, entanglement-spectrum level statistics, and inverse participation ratios. Across both models, increasing circuit depth drives these diagnostics toward random-state-like or random-matrix-like behavior, while variational optimization remains effective, with no evidence of exponential gradient suppression in the regime studied. We interpret this behavior in terms of locality. Spectral correlations develop at relatively shallow depth through locally generated structure, while global state randomization and the associated concentration-of-measure effects are not yet realized. These results show that commonly used statistical diagnostics of complexity do not by themselves determine trainability. Instead, they point to a separation between different aspects of complexity in finite-depth variational circuits.
An efficient representation of a quantum circuit is of great importance in achieving a quantum advantage on current noisy intermediate scale quantum (NISQ) devices and the classical simulation of quantum manybody systems.The quantum circuits are playing the key ingredient in the performance of variational quantum algorithms and quantum dynamics in problems of physics and chemistry.In this paper, we study the role of the network structure of a quantum circuit in its performance.We discuss the variational optimization of quantum circuit (a unitary tensor-network circuit) with different network structures.The ansatz is performed based on a generalization of well-developed multiscale entanglement renormalization algorithm and also the conjugategradient method with an effective line search.We present the benchmarking calculations for different network structures by studying the Heisenberg model in a strongly disordered magnetic field and a tensor-network QR decomposition.Our work can contribute to achieve the most out of NISQ hardware and to classically develop isometric tensor network states.
Parameterized quantum circuits used as variational ansatzes are emerging as promising tools to tackle complex problems ranging from quantum chemistry to combinatorial optimization.These variational quantum circuits can suffer from the well-known curse of barren plateaus, which is characterized by an exponential vanishing of the cost-function gradient with the system size, making training unfeasible for practical applications.Since a generic quantum circuit cannot be simulated efficiently, the determination of its trainability is an important problem.Here we find an efficient method to compute the gradient of the cost function and its variance for a wide class of variational quantum circuits.Our scheme relies on our proof of an exact mapping from randomly initialized circuits to a set of Clifford circuits that can be efficiently simulated on a classical computer by virtue of the celebrated Gottesman-Knill theorem.This method is scalable and can be used to certify trainability for variational quantum circuits and explore design strategies that can overcome the barren-plateau problem.As illustrative examples, we show results with up to 100 qubits.
The exploration of hybrid quantum-classical algorithms and programming models on noisy near-term quantum hardware has begun. As hybrid programs scale towards classical intractability, validation and benchmarking are critical to understanding the utility of the hybrid computational model. In this paper, we demonstrate a newly developed quantum circuit simulator based on tensor network theory that enables intermediate-scale verification and validation of hybrid quantum-classical computing frameworks and programming models. We present our tensor-network quantum virtual machine (TNQVM) simulator which stores a multi-qubit wavefunction in a compressed (factorized) form as a matrix product state, thus enabling single-node simulations of larger qubit registers, as compared to brute-force state-vector simulators. Our simulator is designed to be extensible in both the tensor network form and the classical hardware used to run the simulation (multicore, GPU, distributed). The extensibility of the TNQVM simulator with respect to the simulation hardware type is achieved via a pluggable interface for different numerical backends (e.g., ITensor and ExaTENSOR numerical libraries). We demonstrate the utility of our TNQVM quantum circuit simulator through the verification of randomized quantum circuits and the variational quantum eigensolver algorithm, both expressed within the eXtreme-scale ACCelerator (XACC) programming model.
Whether parameterized quantum circuits (PQCs) can be systematically constructed to be both trainable and expressive remains an open question. Highly expressive PQCs often exhibit barren plateaus, while several trainable alternatives admit efficient classical simulation. We address this question by deriving a finite-sample, dimension-independent concentration bound for estimating the variance of a PQC cost function, yielding explicit trainability guarantees. Across commonly used ansätze, we observe an anticorrelation between trainability and expressibility, consistent with theoretical insights. Building on this observation, we propose a property-based ansatz-search framework for identifying circuits that combine trainability and expressibility. We demonstrate its practical viability on a real quantum computer and apply it to variational quantum algorithms. We identify quantum neural network ansätze with improved effective dimension using over 6×6 \times6× fewer parameters, and for VQE on H2\mathrm{H}_2H2​ we achieve UCCSD-like accuracy at substantially reduced circuit complexity.