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arXiv:2012.10268 · International Journal of Quantum Information

Gate-based circuit designs for quantum adder-inspired quantum random walks on superconducting qubits

Daniel Koch, Michael Samodurov, Andrew M. Projansky, Paul M. Alsing

Published Feb 26, 2022• 4 citations
arXivHTMLPDF

Abstract

Quantum random walks, which drew much attention over the past few decades for their distinctly nonclassical behavior, are a promising subfield within quantum computing. Theoretical framework and applications for these walks have seen many great mathematical advances, with experimental demonstrations now catching up. In this study, we examine the viability of implementing coin quantum random walks using a Quantum Adder-based Shift Operator, with quantum circuit designs specifically for superconducting qubits. We focus on the strengths and weaknesses of these walks, particularly circuit depth, gate count, connectivity requirements and scalability. We propose and analyze a novel approach to implementing boundary conditions for these walks, demonstrating the technique explicitly in one and two dimensions. Finally, we present several fidelity results from running our circuits on IBM’s quantum volume 32 ‘Toronto’ chip, showcasing the extent to which these noisy intermediate-scale quantum devices can currently handle quantum walks.

DOI
10.1142/s021974992150043x

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Quantum random walks, which drew much attention over the past few decades for their distinctly nonclassical behavior, are a promising subfield within quantum computing. Theoretical framework and applications for these walks have seen many great mathematical advances, with experimental demonstrations now catching up. In this study, we examine the viability of implementing coin quantum random walks using a Quantum Adder-based Shift Operator, with quantum circuit designs specifically for superconducting qubits. We focus on the strengths and weaknesses of these walks, particularly circuit depth, gate count, connectivity requirements and scalability. We propose and analyze a novel approach to implementing boundary conditions for these walks, demonstrating the technique explicitly in one and two dimensions. Finally, we present several fidelity results from running our circuits on IBM’s quantum volume 32 ‘Toronto’ chip, showcasing the extent to which these noisy intermediate-scale quantum devices can currently handle quantum walks.
We propose a quantum circuit design for implementing coined quantum walks on complex networks. In complex networks, the coin and shift operators depend on the varying degrees of the nodes, which makes circuit construction more challenging than for regular networks. To address this issue, we use a dual-register encoding to enable a simplified shift operator and reduces the resource overhead. We implement the circuit using Qmod, a high-level quantum programming language, and evaluated the performance through numerical simulations on Erdős-Rényi, Watts-Strogatz, and Barabási-Albert models. The results show that the circuit depth scales as approximately N1.9N^{1.9}N1.9 regardless of the network topology. Furthermore, we execute the proposed circuits on the ibm\_torino superconducting quantum processor for Watts-Strogatz models with N=4N=4N=4 and N=8N=8N=8. The experiments show that hardware-aware optimization slightly improved the variation distance and Hellinger fidelity for the larger network, whereas connectivity constraints imposed overhead for the smaller one. These results indicate that while current NISQ devices are limited to small-scale validations, the polynomial scaling of our framework makes it suitable for larger-scale implementations in the fault-tolerant quantum computing era.
The "Noisy intermediate-scale quantum" NISQ machine era primarily focuses on mitigating noise, controlling errors, and executing high-fidelity operations, hence requiring shallow circuit depth and noise robustness. Approximate computing is a novel computing paradigm that produces imprecise results by relaxing the need for fully precise output for error-tolerant applications including multimedia, data mining, and image processing. We investigate how approximate computing can improve the noise resilience of quantum adder circuits in NISQ quantum computing. We propose five designs of approximate quantum adders to reduce depth while making them noise-resilient, in which three designs are with carryout, while two are without carryout. We have used novel design approaches that include approximating the Sum only from the inputs (pass-through designs) and having zero depth, as they need no quantum gates. The second design style uses a single CNOT gate to approximate the SUM with a constant depth of O(1). We performed our experimentation on IBM Qiskit on noise models including thermal, depolarizing, amplitude damping, phase damping, and bitflip: (i) Compared to exact quantum ripple carry adder without carryout the proposed approximate adders without carryout have improved fidelity ranging from 8.34% to 219.22%, and (ii) Compared to exact quantum ripple carry adder with carryout the proposed approximate adders with carryout have improved fidelity ranging from 8.23% to 371%. Further, the proposed approximate quantum adders are evaluated in terms of various error metrics.
Quantum walks have proven to be a universal model for quantum computation and to provide speed-up in certain quantum algorithms. The discrete-time quantum walk (DTQW) model, among others, is one of the most suitable candidates for circuit implementation due to its discrete nature. Current implementations, however, are usually characterized by quantum circuits of large size and depth, which leads to a higher computational cost and severely limits the number of time steps that can be reliably implemented on current quantum computers. In this work, we propose an efficient and scalable quantum circuit implementing the DTQW on the 2n-cycle based on the diagonalization of the conditional shift operator. For t time steps of the DTQW, the proposed circuit requires only O(n2+nt) two-qubit gates compared to the O(n2t) of the current most efficient implementation based on quantum Fourier transforms. We test the proposed circuit on an IBM quantum device for a Hadamard DTQW on the 4-cycle and 8-cycle characterized by periodic dynamics and by recurrent generation of maximally entangled single-particle states. Experimental results are meaningful well beyond the regime of few time steps, paving the way for reliable implementation and use on quantum computers.
We investigate the counterparts of random walks in universal quantum computing and their implementation using standard quantum circuits. Quantum walks have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1D graph, namely a circle, and show how to implement a discrete-time quantum walk in quantum circuits built with universal CNOT and single qubit gates. We review elementary quantum gates and circuit decomposition techniques and propose a generalized version of all CNOT-based circuits of the quantum walk. We simulated these circuits on five different qubits IBM-Q quantum devices. This quantum computer has nonperfect gates based on superconducting qubits, and, therefore, we analyzed the impact of the CNOT errors and CNOT-depth on the fidelity of the circuit.
Quantum walks (QWs) are of interest as examples of uniquely quantum behavior and are applicable in a variety of quantum search and simulation models. Implementing QWs on quantum devices is useful from both points of view. We describe a prototype one-dimensional QW algorithm that economizes resources required in its implementation. Our algorithm needs only a single shift (increment) operation. It also allows complete flexibility in choosing the shift circuit, a resource intensive part of QW implementations. This is desirable for Noisy Intermediate-Scale Quantum (NISQ) devices, in which fewer computations implies faster execution and reduced effects of noise and decoherence. We implement versions of the algorithm, with two different shift circuit structures, on publicly accessible IBM quantum computers.
Abstract A random walk is one of the widely adopted random processes for simulation and approximation in multiple areas of science and engineering. Quantum random walk is an analog version of the classical random walk. It was first introduced in 1993 by Y. Aharonov et al. They presented that, with the nature of quantum characteristic, the average length of the walking path on a line is possible to larger than a classical random walk can produce. Therefore, the quantum random walk can be used as a tool to construct many other quantum algorithms. Moreover, it can solve graph problems that many real-world problems can be formulated. However, quantum algorithms are only useful in practice if we can implement them efficiently on a quantum computer. In this study, we are then interested in designing and implementing a quantum circuit that can be run on a real quantum computing device. We firstly focus on the most straightforward, which is a one-dimension, quantum random walk algorithm. Then, the quantum circuits are developed and implemented on a real quantum computer and a quantum computing simulator using software development kits provided by IBM. Lastly, the performance and results of the quantum circuits tested on both computing platforms are presented.