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Noisy intermediate-scale quantum (NISQ) algorithms

Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, Alán Aspuru-Guzik

TL;DR

A thorough summary of NISQ computational paradigms and algorithms, which discusses the key structure of these algorithms, their limitations, and advantages, and a comprehensive overview of various benchmarking and software tools useful for programming and testing NISZ devices.

Abstract

A universal fault-tolerant quantum computer that can solve efficiently problems such as integer factorization and unstructured database search requires millions of qubits with low error rates and long coherence times. While the experimental advancement towards realizing such devices will potentially take decades of research, noisy intermediate-scale quantum (NISQ) computers already exist. These computers are composed of hundreds of noisy qubits, i.e. qubits that are not error-corrected, and therefore perform imperfect operations in a limited coherence time. In the search for quantum advantage with these devices, algorithms have been proposed for applications in various disciplines spanning physics, machine learning, quantum chemistry and combinatorial optimization. The goal of such algorithms is to leverage the limited available resources to perform classically challenging tasks. In this review, we provide a thorough summary of NISQ computational paradigms and algorithms. We discuss the key structure of these algorithms, their limitations, and advantages. We additionally provide a comprehensive overview of various benchmarking and software tools useful for programming and testing NISQ devices.

Noisy intermediate-scale quantum (NISQ) algorithms

TL;DR

A thorough summary of NISQ computational paradigms and algorithms, which discusses the key structure of these algorithms, their limitations, and advantages, and a comprehensive overview of various benchmarking and software tools useful for programming and testing NISZ devices.

Abstract

A universal fault-tolerant quantum computer that can solve efficiently problems such as integer factorization and unstructured database search requires millions of qubits with low error rates and long coherence times. While the experimental advancement towards realizing such devices will potentially take decades of research, noisy intermediate-scale quantum (NISQ) computers already exist. These computers are composed of hundreds of noisy qubits, i.e. qubits that are not error-corrected, and therefore perform imperfect operations in a limited coherence time. In the search for quantum advantage with these devices, algorithms have been proposed for applications in various disciplines spanning physics, machine learning, quantum chemistry and combinatorial optimization. The goal of such algorithms is to leverage the limited available resources to perform classically challenging tasks. In this review, we provide a thorough summary of NISQ computational paradigms and algorithms. We discuss the key structure of these algorithms, their limitations, and advantages. We additionally provide a comprehensive overview of various benchmarking and software tools useful for programming and testing NISQ devices.

Paper Structure

This paper contains 179 sections, 1 theorem, 143 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Computational complexity theory in a nutshell
  3. Experimental progress
  4. NISQ and near-term
  5. Scope of the review
  6. Building blocks of variational quantum algorithms
  7. Building blocks of variational quantum algorithms
  8. Objective function
  9. Pauli strings
  10. Fidelity
  11. Other objective functions
  12. Parameterized quantum circuits
  13. Problem-inspired ansätze
  14. Unitary Coupled Cluster.
  15. Factorized Unitary Coupled-Cluster and Adaptive Approaches.
  16. ...and 164 more sections

Key Result

Lemma 6

Given a finite set $S\subset \mathbb{U}(d)$ and a weight function $w:S\rightarrow(0,1]$, the tuple$(S,w)$ forms a $t$-design if and only if $\sum_{U\in S}w(U)U^{\otimes t}\otimes\left(U^{\dagger}\right)^{\otimes t}=\int_{U(d)}U^{\otimes t}\otimes\left(U^{\dagger}\right)^{\otimes t}dU.$

Figures (7)

  • Figure 1: An illustrative picture of some relevant complexity classes together with a problem examples. For the chess example, the word "restricted" refers to a polynomial upper bound on the number of moves. The containment relations are suggestive. Some of them have not been mathematically proven, being a well-known open problem whether $\textrm{P}$ is equal to $\textrm{NP}$.
  • Figure 2: Diagrammatic representation of a Variational Quantum Algorithm (VQA). A VQA workflow can be divided into four main components: a) the objective function $O$ that encodes the problem to be solved; b) the parameterized quantum circuit (PQC) $U$, which variables $\boldsymbol{\theta}$ are tuned to minimize the objective; c) the measurement scheme, which performs the basis changes and measurements needed to compute expectation values that are used to evaluate the objective; and d) the classical optimizer that minimizes the objective. The PQC can be defined heuristically, following hardware-inspired ansätze, or designed from the knowledge about the problem Hamiltonian $H$. Inputs of a VQA are the circuit ansatz $U(\boldsymbol\theta)$ and the initial parameter values $\boldsymbol\theta_0$. Outputs include optimized parameter values $\boldsymbol{\theta}^*$and the minimum of the objective.
  • Figure 3: Example problem-inspired and hardware-efficient ansätze. (a) Circuit of the Unitary Coupled Cluster ansatz with a detailed view of a fermionic excitation as discussed in yordanov2020efficient. (b) Hardware-efficient ansatz tailored to a processor that is optimized for single-qubit $x$- and $z$-rotations and nearest-neighbor two-qubit CNOT gates.
  • Figure 4: Gaussian boson sampling circuit for a photonic setup. The qumodes are prepared in gaussian states from the vacuum by squeezing operations $S(z_i)$, followed by an interferometer consisting of phaseshifters $R(\theta) = e^{i\theta_j}$ and beam-splitters BS. At the end, photon number resolving measurements are made in each mode.
  • Figure 5: Quantum computing of the expected value of an observable using gate set tomography-based PEC. Quasiprobability decomposition of initial state preparation, examplary single- and two-qubit processes are computed. Implementing the resulting decomposition is done using the Monte Carlo approach. With QEM, the probability distribution of expected value of the physical observable is now centered around an ideal value with larger variance as compared to the one without QEM. Inspired by zhang2020error.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 6