scholarly journals On the universality of the quantum approximate optimization algorithm

2020 ◽  
Vol 19 (9) ◽  
Author(s):  
M. E. S. Morales ◽  
J. D. Biamonte ◽  
Z. Zimborás
Keyword(s):  
Optimization Algorithm ◽  
Quantum Circuit ◽  
Original Form ◽  
Complete Proof ◽  
One Dimensional ◽  
Approximate Optimization ◽  
Near Term ◽  
Set Up ◽  
The Cost ◽  
Universal Gate

Abstract The quantum approximate optimization algorithm (QAOA) is considered to be one of the most promising approaches towards using near-term quantum computers for practical application. In its original form, the algorithm applies two different Hamiltonians, called the mixer and the cost Hamiltonian, in alternation with the goal being to approach the ground state of the cost Hamiltonian. Recently, it has been suggested that one might use such a set-up as a parametric quantum circuit with possibly some other goal than reaching ground states. From this perspective, a recent work (Lloyd, arXiv:1812.11075) argued that for one-dimensional local cost Hamiltonians, composed of nearest neighbour ZZ terms, this set-up is quantum computationally universal and provides a universal gate set, i.e. all unitaries can be reached up to arbitrary precision. In the present paper, we complement this work by giving a complete proof and the precise conditions under which such a one-dimensional QAOA might produce a universal gate set. We further generalize this type of gate-set universality for certain cost Hamiltonians with ZZ and ZZZ terms arranged according to the adjacency structure of certain graphs and hypergraphs.

scholarly journals Quantum Approximate Optimization Algorithm: Performance, Mechanism, and Implementation on Near-Term Devices

2020 ◽  
Vol 10 (2) ◽  
Author(s):  
Leo Zhou ◽  
Sheng-Tao Wang ◽  
Soonwon Choi ◽  
Hannes Pichler ◽  
Mikhail D. Lukin
Keyword(s):  
Optimization Algorithm ◽  
Algorithm Performance ◽  
Approximate Optimization ◽  
Near Term

scholarly journals Modelling of Grover’s quantum search algorithms: implementations of Simple quantum simulators on classical computers

2020 ◽  
pp. 65-128
Author(s):  
Sergey Ulyanov ◽  
Andrey Reshetnikov ◽  
Olga Tyatyushkina
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Search Algorithm ◽  
Quantum Circuit ◽  
Unitary Matrix ◽  
Single Qubit ◽  
Simple Quantum ◽  
Grover’S Search Algorithm ◽  
The Cost ◽  
Universal Gate

Models of Grover’s search algorithm is reviewed to build the foundation for the other algorithms. Thereafter, some preliminary modifications of the original algorithms by others are stated, that increases the applicability of the search procedure. A general quantum computation on an isolated system can be represented by a unitary matrix. In order to execute such a computation on a quantum computer, it is common to decompose the unitary into a quantum circuit, i.e., a sequence of quantum gates that can be physically implemented on a given architecture. There are different universal gate sets for quantum computation. Here we choose the universal gate set consisting of CNOT and single-qubit gates. We measure the cost of a circuit by the number of CNOT gates as they are usually more difficult to implement than single qubit gates and since the number of single-qubit gates is bounded by about twice the number of CNOT’s.

scholarly journals Classical variational simulation of the Quantum Approximate Optimization Algorithm

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Matija Medvidović ◽  
Giuseppe Carleo
Keyword(s):  
Optimization Algorithm ◽  
Large Scale ◽  
Quantum Algorithms ◽  
Practical Interest ◽  
Approximate Optimization ◽  
Near Term ◽  
Classical Computing ◽  
The Many ◽  
Computational Resources ◽  
Parameter Values

AbstractA key open question in quantum computing is whether quantum algorithms can potentially offer a significant advantage over classical algorithms for tasks of practical interest. Understanding the limits of classical computing in simulating quantum systems is an important component of addressing this question. We introduce a method to simulate layered quantum circuits consisting of parametrized gates, an architecture behind many variational quantum algorithms suitable for near-term quantum computers. A neural-network parametrization of the many-qubit wavefunction is used, focusing on states relevant for the Quantum Approximate Optimization Algorithm (QAOA). For the largest circuits simulated, we reach 54 qubits at 4 QAOA layers, approximately implementing 324 RZZ gates and 216 RX gates without requiring large-scale computational resources. For larger systems, our approach can be used to provide accurate QAOA simulations at previously unexplored parameter values and to benchmark the next generation of experiments in the Noisy Intermediate-Scale Quantum (NISQ) era.

scholarly journals Temporal Planning for Compilation of Quantum Approximate Optimization Circuits

2017 ◽  
Author(s):  
Davide Venturelli ◽  
Minh Do ◽  
Eleanor Rieffel ◽  
Jeremy Frank
Keyword(s):  
Optimization Algorithm ◽  
Empirical Evaluation ◽  
Quantum Circuit ◽  
Hardware Architecture ◽  
Quantum Circuits ◽  
Approximate Optimization ◽  
Temporal Planning ◽  
Ordering Constraints ◽  
Viable Approach

We investigate the application of temporal planners to the problem of compiling quantum circuits to emerging quantum hardware. While our approach is general, we focus our initial experiments on Quantum Approximate Optimization Algorithm (QAOA) circuits that have few ordering constraints and thus allow highly parallel plans. We report on experiments using several temporal planners to compile circuits of various sizes to a realistic hardware architecture. This early empirical evaluation suggests that temporal planning is a viable approach to quantum circuit compilation.

Quantum Circuit Compilation by Genetic Algorithm for Quantum Approximate Optimization Algorithm applied to MaxCut Problem

2022 ◽  
pp. 101030
Author(s):  
Lis Arufe ◽  
Miguel A. González ◽  
Angelo Oddi ◽  
Riccardo Rasconi ◽  
Ramiro Varela
Keyword(s):  
Genetic Algorithm ◽  
Optimization Algorithm ◽  
Quantum Circuit ◽  
Approximate Optimization ◽  
Maxcut Problem

scholarly journals Quantum annealing initialization of the quantum approximate optimization algorithm

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 491
Author(s):  
Stefan H. Sack ◽  
Maksym Serbyn
Keyword(s):  
Optimization Algorithm ◽  
Quantum Algorithm ◽  
External Parameter ◽  
Quantum Annealing ◽  
Time Step ◽  
Quantum Devices ◽  
Approximate Optimization ◽  
Optimal Value ◽  
Random Initialization ◽  
Near Term

The quantum approximate optimization algorithm (QAOA) is a prospective near-term quantum algorithm due to its modest circuit depth and promising benchmarks. However, an external parameter optimization required in QAOA could become a performance bottleneck. This motivates studies of the optimization landscape and search for heuristic ways of parameter initialization. In this work we visualize the optimization landscape of the QAOA applied to the MaxCut problem on random graphs, demonstrating that random initialization of the QAOA is prone to converging to local minima with sub-optimal performance. We introduce the initialization of QAOA parameters based on the Trotterized quantum annealing (TQA) protocol, parameterized by the Trotter time step. We find that the TQA initialization allows to circumvent the issue of false minima for a broad range of time steps, yielding the same performance as the best result out of an exponentially scaling number of random initializations. Moreover, we demonstrate that the optimal value of the time step coincides with the point of proliferation of Trotter errors in quantum annealing. Our results suggest practical ways of initializing QAOA protocols on near-term quantum devices and reveals new connections between QAOA and quantum annealing.

scholarly journals Lower bounds on circuit depth of the quantum approximate optimization algorithm

2021 ◽  
Vol 20 (2) ◽  
Author(s):  
Rebekah Herrman ◽  
James Ostrowski ◽  
Travis S. Humble ◽  
George Siopsis
Keyword(s):  
Lower Bounds ◽  
Optimization Algorithm ◽  
Approximate Optimization ◽  
Circuit Depth

scholarly journals A framework for reducing the overhead of the quantum oracle for use with Grover’s algorithm with applications to cryptanalysis of SIKE

2020 ◽  
Vol 15 (1) ◽  
pp. 143-156
Author(s):  
Jean-François Biasse ◽  
Benjamin Pring
Keyword(s):  
Search Algorithm ◽  
Circuit Complexity ◽  
Quantum Circuit ◽  
Grover’S Algorithm ◽  
Search Problems ◽  
Trade Offs ◽  
Single Target ◽  
Grover's Algorithm ◽  
Quantum Oracle ◽  
The Cost

AbstractIn this paper we provide a framework for applying classical search and preprocessing to quantum oracles for use with Grover’s quantum search algorithm in order to lower the quantum circuit-complexity of Grover’s algorithm for single-target search problems. This has the effect (for certain problems) of reducing a portion of the polynomial overhead contributed by the implementation cost of quantum oracles and can be used to provide either strict improvements or advantageous trade-offs in circuit-complexity. Our results indicate that it is possible for quantum oracles for certain single-target preimage search problems to reduce the quantum circuit-size from $O\left(2^{n/2}\cdot mC\right)$ (where C originates from the cost of implementing the quantum oracle) to $O(2^{n/2} \cdot m\sqrt{C})$ without the use of quantum ram, whilst also slightly reducing the number of required qubits.This framework captures a previous optimisation of Grover’s algorithm using preprocessing [21] applied to cryptanalysis, providing new asymptotic analysis. We additionally provide insights and asymptotic improvements on recent cryptanalysis [16] of SIKE [14] via Grover’s algorithm, demonstrating that the speedup applies to this attack and impacting upon quantum security estimates [16] incorporated into the SIKE specification [14].

scholarly journals A trade-off between classical and quantum circuit size for an attack against CSIDH

2020 ◽  
Vol 15 (1) ◽  
pp. 4-17
Author(s):  
Jean-François Biasse ◽  
Xavier Bonnetain ◽  
Benjamin Pring ◽  
André Schrottenloher ◽  
William Youmans
Keyword(s):  
Endomorphism Ring ◽  
State Of The Art ◽  
Quantum Circuit ◽  
List Type ◽  
Hard Problem ◽  
Trade Off ◽  
Classical State ◽  
Security Parameter ◽  
The Cost ◽  
Quadratic Order

AbstractWe propose a heuristic algorithm to solve the underlying hard problem of the CSIDH cryptosystem (and other isogeny-based cryptosystems using elliptic curves with endomorphism ring isomorphic to an imaginary quadratic order 𝒪). Let Δ = Disc(𝒪) (in CSIDH, Δ = −4p for p the security parameter). Let 0 < α < 1/2, our algorithm requires:A classical circuit of size $2^{\tilde{O}\left(\log(|\Delta|)^{1-\alpha}\right)}.$A quantum circuit of size $2^{\tilde{O}\left(\log(|\Delta|)^{\alpha}\right)}.$Polynomial classical and quantum memory.Essentially, we propose to reduce the size of the quantum circuit below the state-of-the-art complexity $2^{\tilde{O}\left(\log(|\Delta|)^{1/2}\right)}$ at the cost of increasing the classical circuit-size required. The required classical circuit remains subexponential, which is a superpolynomial improvement over the classical state-of-the-art exponential solutions to these problems. Our method requires polynomial memory, both classical and quantum.

A Determinant Elimination Method for Bottleneck Assignment and Generalized Assignment Problems

2012 ◽  
Vol 239-240 ◽  
pp. 1522-1527
Author(s):  
Wen Bo Wu ◽  
Yu Fu Jia ◽  
Hong Xing Sun
Keyword(s):  
Computational Complexity ◽  
Optimization Algorithm ◽  
Assignment Problem ◽  
Numerical Experiments ◽  
High Efficiency ◽  
Synthesis Method ◽  
Assignment Problems ◽  
Generalized Assignment ◽  
The Cost ◽  
Time And Space Complexity

The bottleneck assignment (BA) and the generalized assignment (GA) problems and their exact solutions are explored in this paper. Firstly, a determinant elimination (DE) method is proposed based on the discussion of the time and space complexity of the enumeration method for both BA and GA problems. The optimization algorithm to the pre-assignment problem is then discussed and the adjusting and transformation to the cost matrix is adopted to reduce the computational complexity of the DE method. Finally, a synthesis method for both BA and GA problems is presented. The numerical experiments are carried out and the results indicate that the proposed method is feasible and of high efficiency.

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