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.
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.
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.
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.
The viability of algorithms on Noisy Intermediate-Scale Quantum (NISQ) computers is limited by operational errors and decoherence. To optimize execution, it is crucial to understand how circuit complexity (depth and number of gates), statistical sampling (shots), and result fidelity interact. This work investigates that relationship through large-scale simulations of 100,000 random circuits under a physically motivated noise model with temporal decoherence (T1/T2) and gate errors. The analysis shows a strong negative correlation between circuit complexity and Hellinger fidelity, revealing a “fidelity ceiling” where increasing shots reduces statistical error but cannot overcome noise. For complex circuits, the number of gates is identified as a stronger predictor of error than circuit depth.
Noisy intermediate-scale quantum (NISQ) devices pave the way for implementing quantum algorithms that offer quantum advantages over their classical counterparts. Due to the intrinsic noise and decoherence in the physical system, NISQ machines are naturally modeled as large-scale, low-depth quantum circuits. In practice, executing such circuits requires sending program states that encode the relevant instructions to a programmable quantum computer, typically through a cloud service. Existing programming approaches designed for generic unitary transformations are computationally inefficient in the low-depth setting, and therefore remain unsatisfactory. As such, to realize NISQ algorithms, it is crucial to find an efficient way to program low-depth circuits as the number of qubits N increases. Here, we investigate the circuit complexity and the size of quantum memory, known as the program cost, required to program low-depth brickwork circuits. We establish a tight worst-case program cost of Θ ( N p o l y l o g N ) for universally programming low-depth brickwork circuits in the large- N regime. Moreover, we analyze the trade-off between the cost of describing the layout of local gates and the cost of programming them to implement the target unitaries via the light-cone argument. Our findings suggest that faithful gate-wise programming is essentially optimal in the low-depth regime.
Abstract Efficient computation of molecular energies is an exciting application of quantum computing for quantum chemistry, but current noisy intermediate-scale quantum (NISQ) devices can only execute shallow circuits, limiting existing variational quantum algorithms, which require deep entangling quantum circuit ansatzes to capture correlations, to small molecules. Here we demonstrate a variational NISQ-friendly algorithm that generates a set of Hartree–Fock ansatzes using multiple shallow circuits with depth linear in the number of qubits to estimate electronic correlation energies via perturbation theory up to the second order. We tested the algorithm on several small molecules, both with classical simulations including noise models and on cloud quantum processors, showing that it not only reproduces the equilibrium molecular energies but it also captures the perturbative electronic correlation effects at longer bond distances. The number of shallow circuits required for our algorithms scales O ( N 5 ) , with N being the system size, thus enabling the study of larger molecules compared to other approaches, where circuit depth or number of qubits required become prohibitive for NISQ devices.
Understanding the computational power of noisy intermediate-scale quantum (NISQ) devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. Specifically, we study noisy random circuit sampling in one dimension (or 1D noisy RCS) as a simple model for exploring the effects of noise on the computational power of a noisy quantum device. In particular, we simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. The latter metric is chosen because it determines the cost of classical MPO simulation. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. Specifically, we show that above the characteristic system size, there is an optimal circuit depth, independent of the system size, where the MPO entanglement entropy is maximized. Most importantly, the maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size. We also provide a heuristic analysis to get the scaling of the maximum achievable MPO entanglement entropy as a function of the gate error rate. The obtained scaling suggests that although the cost of MPO simulation does not increase exponentially in the system size above a certain characteristic system size, it does increase exponentially as the gate error rate decreases, possibly making classical simulation practically not feasible even with state-of-the-art supercomputers.