One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, this tradeoff still limits the size of tractable problems since the increased depth is often not realizable before noise dominates. Here, we develop qubit-efficient quantum algorithms for entanglement spectroscopy which avoid this tradeoff. In particular, we develop algorithms for computing the trace of the n -th power of the density operator of a quantum system, T r ( ρ n ) , (related to the Rényi entropy of order n ) that use fewer qubits than any previous efficient algorithm while achieving similar performance in the presence of noise, thus enabling spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which require a number of qubits independent of n , are variants of previous algorithms with width proportional to n , an asymptotic difference. The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation, allowing us to reuse qubits and increase the circuit depth without suffering the usual noisy consequences. We also introduce the notion of effective circuit depth as a generalization of standard circuit depth suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms and should aid in designing future algorithms. We perform numerical simulations to compare our algorithms to the original variants and show they perform similarly when subjected to noise. Additionally, we experimentally implement one of our qubit-efficient algorithms on the Honeywell System Model H0, estimating T r ( ρ n ) for larger n than possible with previous algorithms.
One strategy to fit larger problems on NISQ devices is to exploit a tradeoff between circuit width and circuit depth. Unfortunately, this tradeoff still limits the size of tractable problems since the increased depth is often not realizable before noise dominates. Here, we develop qubit-efficient quantum algorithms for entanglement spectroscopy which avoid this tradeoff. In particular, we develop algorithms for computing the trace of the n -th power of the density operator of a quantum system, T r ( ρ n ) , (related to the Rényi entropy of order n ) that use fewer qubits than any previous efficient algorithm while achieving similar performance in the presence of noise, thus enabling spectroscopy of larger quantum systems on NISQ devices. Our algorithms, which require a number of qubits independent of n , are variants of previous algorithms with width proportional to n , an asymptotic difference. The crucial ingredient in these new algorithms is the ability to measure and reinitialize subsets of qubits in the course of the computation, allowing us to reuse qubits and increase the circuit depth without suffering the usual noisy consequences. We also introduce the notion of effective circuit depth as a generalization of standard circuit depth suitable for circuits with qubit resets. This tool helps explain the noise-resilience of our qubit-efficient algorithms and should aid in designing future algorithms. We perform numerical simulations to compare our algorithms to the original variants and show they perform similarly when subjected to noise. Additionally, we experimentally implement one of our qubit-efficient algorithms on the Honeywell System Model H0, estimating T r ( ρ n ) for larger n than possible with previous algorithms.
Abstract Noisy intermediate-scale quantum (NISQ) computers have gate errors and decoherence, limiting the depth of circuits that can be implemented on them. A strategy for NISQ algorithms is to reduce the circuit depth at the expense of increasing the qubit count. Here, we exploit this trade-off for an application called entanglement spectroscopy, where one computes the entanglement of a state on systems by evaluating the Rényi entropy of the reduced state . For a -qubit state , the Rényi entropy of order is computed via , with the complexity growing exponentially in for classical computers. Johri et al (2017 Phys. Rev . B 96 195136) introduced a quantum algorithm that requires copies of and whose depth scales linearly in . Here, we present a quantum algorithm requiring twice the qubit resources ( copies of ) but with a depth that is independent of both and . Surprisingly this depth is only two gates. Our numerical simulations show that this short depth leads to an increased robustness to noise.
The advent of noisy intermediate-scale quantum (NISQ) devices has not only opened up new possibilities to studying quantum computational benefits, but it has also presented new challenges associated with noise and short coherence times. Entanglement is one of the inherent resources that can be used to perform quantum computation and implement the performance and reliability of quantum algorithms. This paper looks at entanglement dynamics in NISQ machines subjected to natural noises. The study focuses on the problem of examining the role of various noise channels in the creation, dynamics, and decay of the entanglement during quantum operations, such as depolarizing, amplitude damping, and phase damping. The study is based on theoretical modeling and numerical simulations of multi-qubit quantum circuits to determine the stability of entangled states with respect to circuit depth and intensity of noise. Concurrence and von Neumann entropy are examples of entanglement measures that are used to characterize correlations between qubits as well as to trace the dynamics of these correlations across computational interactions. The results show that noise has major impacts on entanglement distribution, which tends to cause rapid decays in highly entangled states with increase in circuit complexity. The findings, however, also show that there are some circuit architectures and error-mitigation schemes that can somewhat sustain entanglement with moderate levels of noise. The paper also emphasizes the relevance of minimalizing sequence of gates, and reduced circuit depth in an effort to preserve useful entanglement in present-day quantum hardware. These observations help in understanding the behaviour of quantum correlations better in real world quantum devices and provide a way of developing more resilient quantum algorithms that would work on NISQ systems. In general, the article offers a systematic view on entanglement behavior in disordered quantum systems and highlights its value in the development of quantum computing applications in the near term.
Circuit cutting is a promising technique for mapping a quantum circuit requiring many qubits to a series of circuits requiring fewer qubits. At the cost of a longer runtime, such a strategy can reduce qubit requirements and mitigate noise in currently available Noisy Intermediate-Scale Quantum (NISQ) devices where noise scales poorly with greater levels of entanglement. In this work, we investigate the performance of wire-based circuit cutting in cases of different levels of entanglement and forms of noise. As a broadly applicable starting point, we focus on cutting Hamiltonian-inspired quantum circuits with various entanglement strategies and circuit depths. We provide model calculations of these circuits before and after cutting, both in the ideal case and in the presence of noise, using quantum simulators with quantum noise models. We present a numerical analysis of the performance of circuit cutting using various metrics for comparing ideal and noisy model calculations. Our results indicate that the efficacy of circuit cutting is sensitive to the level of entanglements and is affected differently by incoherent quantum noises.
Estimating the trace of quantum state powers, Tr ( ρ k ) , for k identical quantum states is a fundamental task with numerous applications in quantum information processing, including nonlinear function estimation of quantum states and entanglement detection. On near-term quantum devices, reducing the required quantum circuit depth, the number of multi-qubit quantum operations, and the copies of the quantum state needed for such computations is crucial. In this work, inspired by the Newton-Girard method, we significantly improve upon existing results by introducing an algorithm that requires only O ( r ~ ) qubits and O ( r ~ ) multi-qubit gates, where r ~ = min { rank ( ρ ) , ⌈ ln ( 2 k / ϵ ) ⌉ } . This approach is efficient, as it employs the r ~ -entangled copy measurement instead of the conventional k -entangled copy measurement, while asymptotically preserving the known sample complexity upper bound. Furthermore, we prove that estimating { Tr ( ρ i ) } i = 1 r ~ is sufficient to approximate Tr ( ρ k ) even for large integers k > r ~ . This leads to a rank-dependent complexity for solving the problem, providing an efficient algorithm for low-rank quantum states while also improving existing methods when the rank is unknown or when the state is not low-rank. Building upon these advantages, we extend our algorithm to the estimation of Tr ( M ρ k ) for arbitrary observables and Tr ( ρ k σ l ) for multiple quantum states.
The ability for users to access quantum computers through the cloud has increased rapidly in recent years. Despite still being Noisy Intermediate-Scale Quantum (NISQ) machines, modern quantum computers are now being actively employed for research and by numerous startups. Quantum algorithms typically produce probabilistic results, necessitating repeated execution to produce the desired outcomes. In order for the execution to begin from the specified ground state each time and for the results of the prior execution not to interfere with the results of the subsequent execution, the reset mechanism must be performed between each iteration to effectively reset the qubits. However, due to noise and errors in quantum computers and specifically these reset mechanisms, a noisy reset operation may lead to systematic errors in the overall computation, as well as potential security and privacy vulnerabilities of information leakage. To counter this issue, we thoroughly examine the state leakage problem in quantum computing, and then propose a solution by employing the classical and quantum one-time pads before the reset mechanism to prevent the state leakage, which works by randomly applying simple gates for each execution of the circuit. In addition, this work explores conditions under which the classical one-time pad, which uses fewer resources, is sufficient to protect state leakage. Finally, we study the role of various errors in state leakage, by evaluating the degrees of leakage under different error levels of gate, measurement, and sampling errors. Our findings offer new perspectives on the design of reset mechanisms and secure quantum computing systems.
Abstract Noisy Intermediate-Scale Quantum (NISQ) devices are near-term quantum computing devices that inherently suffer from imperfect gate implementation, decoherence, and measurement noise. The main type of variational quantum algorithms(VQAs) is based on Parameterized Quantum Circuits (PQCs), which, however, face practical challenges due to optimization instability of optimization caused by noise and reduced performance. In this paper, we suggest a comprehensive noise-resilient optimization framework for PQCs, which explicitly incorporates noise-resilient circuit architecture design, incorporating noise-aware training strategies into the variational optimization loop. The framework uses circuit structures that are efficient in depth and are hardware compatible in addition to adaptive optimizer choice, stochastic optimization, and averaging as a part of repeated measurement to reduce the impacts of realistic noise. Large-scale experiments are performed with different amounts of qubits, depths of the circuit and noise strengths with realistic noise models. The findings indicate that the convergence rate is faster, the stability is increased, the variability of run-to-run variability is minimized, and other robustness is better than the noise-unaware optimization techniques. Moreover, systematic studies determine optimal operating regions that optimally trade-off expressibility and noise tolerance to offer useful design advice on trusted implementation of VQAs with near-term quantum hardware.