scholarly journals Quantum circuits and Spin(3n) groups

2015 ◽  
Vol 15 (3&4) ◽  
pp. 235-259
Author(s):  
Alexander Yu. Vlasov
Keyword(s):  
Tensor Product ◽  
Unitary Transformation ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Spin Groups ◽  
Classical Simulation ◽  
The Matrix ◽  
Universal Quantum ◽  
Usual Quantum

All quantum gates with one and two qubits may be described by elements of Spin groups due to isomorphisms Spin(3)\isomSU(2) and Spin(6)\isomSU(4). However, the group of n-qubit gates SU(2^n) for n>2 has bigger dimension than Spin(3n). A quantum circuit with one- and two-qubit gates may be used for construction of arbitrary unitary transformation SU(2^n). Analogously, the `$Spin(3n)$ circuits' are introduced in this work as products of elements associated with one- and two-qubit gates with respect to the above-mentioned isomorphisms. The matrix tensor product implementation of the Spin(3n) group together with relevant models by usual quantum circuits with 2n qubits are investigated in such a framework. A certain resemblance with well-known sets of non-universal quantum gates (e.g., matchgates, noninteracting-fermion quantum circuits) related with Spin(2n) may be found in presented approach. Finally, a possibility of the classical simulation of such circuits in polynomial time is discussed.

scholarly journals Connectivity matrix model of quantum circuits and its application to distributed quantum circuit optimization

2021 ◽  
Vol 20 (7) ◽  
Author(s):  
Ismail Ghodsollahee ◽  
Zohreh Davarzani ◽  
Mariam Zomorodi ◽  
Paweł Pławiak ◽  
Monireh Houshmand ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Connectivity Matrix ◽  
Circuit Optimization ◽  
Novel Approach ◽  
The Matrix ◽  
Minimum Number ◽  
Two Phases

AbstractAs quantum computation grows, the number of qubits involved in a given quantum computer increases. But due to the physical limitations in the number of qubits of a single quantum device, the computation should be performed in a distributed system. In this paper, a new model of quantum computation based on the matrix representation of quantum circuits is proposed. Then, using this model, we propose a novel approach for reducing the number of teleportations in a distributed quantum circuit. The proposed method consists of two phases: the pre-processing phase and the optimization phase. In the pre-processing phase, it considers the bi-partitioning of quantum circuits by Non-Dominated Sorting Genetic Algorithm (NSGA-III) to minimize the number of global gates and to distribute the quantum circuit into two balanced parts with equal number of qubits and minimum number of global gates. In the optimization phase, two heuristics named Heuristic I and Heuristic II are proposed to optimize the number of teleportations according to the partitioning obtained from the pre-processing phase. Finally, the proposed approach is evaluated on many benchmark quantum circuits. The results of these evaluations show an average of 22.16% improvement in the teleportation cost of the proposed approach compared to the existing works in the literature.

scholarly journals Quantum Implementation of Powell’s Conjugate Direction Method

2019 ◽  
Vol 23 (4) ◽  
pp. 726-734
Author(s):  
Kehan Chen ◽  
Fei Yan ◽  
Kaoru Hirota ◽  
Jianping Zhao ◽  
◽  
...  
Keyword(s):  
Search Algorithm ◽  
Quantum Circuit ◽  
Data Fitting ◽  
Quadratic Equation ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Circuit Implementation ◽  
One Dimensional ◽  
Conjugate Direction ◽  
Conjugate Direction Method

A quantum circuit implementation of Powell’s conjugate direction method (“Powell’s method”) is proposed based on quantum basic transformations in this study. Powell’s method intends to find the minimum of a function, including a sequence of parameters, by changing one parameter at a time. The quantum circuits that implement Powell’s method are logically built by combining quantum computing units and basic quantum gates. The main contributions of this study are the quantum realization of a quadratic equation, the proposal of a quantum one-dimensional search algorithm, the quantum implementation of updating the searching direction array (SDA), and the quantum judgment of stopping the Powell’s iteration. A simulation demonstrates the execution of Powell’s method, and future applications, such as data fitting and image registration, are discussed.

Analyzing Heuristic-based Randomized Search Strategies for the Quantum Circuit Compilation Problem

2020 ◽  
Vol 174 (3-4) ◽  
pp. 259-281
Author(s):  
Angelo Oddi ◽  
Riccardo Rasconi
Keyword(s):  
Quantum Computing ◽  
Nearest Neighbor ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Ranking Functions ◽  
Solution Quality ◽  
Circuit Realization ◽  
Definition Of

In this work we investigate the performance of greedy randomised search (GRS) techniques to the problem of compiling quantum circuits to emerging quantum hardware. Quantum computing (QC) represents the next big step towards power consumption minimisation and CPU speed boost in the future of computing machines. Quantum computing uses quantum gates that manipulate multi-valued bits (qubits). A quantum circuit is composed of a number of qubits and a series of quantum gates that operate on those qubits, and whose execution realises a specific quantum algorithm. Current quantum computing technologies limit the qubit interaction distance allowing the execution of gates between adjacent qubits only. This has opened the way to the exploration of possible techniques aimed at guaranteeing nearest-neighbor (NN) compliance in any quantum circuit through the addition of a number of so-called swap gates between adjacent qubits. In addition, technological limitations (decoherence effect) impose that the overall duration (makespan) of the quantum circuit realization be minimized. One core contribution of the paper is the definition of two lexicographic ranking functions for quantum gate selection, using two keys: one key acts as a global closure metric to minimise the solution makespan; the second one is a local metric, which favours the mutual approach of the closest qstates pairs. We present a GRS procedure that synthesises NN-compliant quantum circuits realizations, starting from a set of benchmark instances of different size belonging to the Quantum Approximate Optimization Algorithm (QAOA) class tailored for the MaxCut problem. We propose a comparison between the presented meta-heuristics and the approaches used in the recent literature against the same benchmarks, both from the CPU efficiency and from the solution quality standpoint. In particular, we compare our approach against a reference benchmark initially proposed and subsequently expanded in [1] by considering: (i) variable qubit state initialisation and (ii) crosstalk constraints that further restrict parallel gate execution.

scholarly journals Matchgates and classical simulation of quantum circuits

2008 ◽  
Vol 464 (2100) ◽  
pp. 3089-3106 ◽  
Author(s):  
Richard Jozsa ◽  
Akimasa Miyake
Keyword(s):  
Clifford Algebra ◽  
Point Of View ◽  
Quantum Circuits ◽  
Nearest Neighbour ◽  
Computational Power ◽  
Classical Simulation ◽  
Wigner Representation ◽  
Universal Quantum ◽  
The One ◽  
Qubit Gate

Let G ( A ,  B ) denote the two-qubit gate that acts as the one-qubit SU (2) gates A and B in the even and odd parity subspaces, respectively, of two qubits. Using a Clifford algebra formalism, we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allow the same gates to act only on n.n. and next n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan–Wigner representation of a Clifford algebra, we show how one may generalize the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.

scholarly journals Towards optimization of quantum circuits

Open Physics ◽  
2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Michal Sedlák ◽  
Martin Plesch
Keyword(s):  
Quantum Information ◽  
Quantum Information Processing ◽  
Quantum Circuit ◽  
Quantum Gate ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Unitary Operation ◽  
Short Sequence ◽  
Toffoli Gates ◽  
Universal Procedure

AbstractAny unitary operation in quantum information processing can be implemented via a sequence of simpler steps — quantum gates. However, actual implementation of a quantum gate is always imperfect and takes a finite time. Therefore, searching for a short sequence of gates — efficient quantum circuit for a given operation, is an important task. We contribute to this issue by proposing optimization of the well-known universal procedure proposed by Barenco et al. [Phys. Rev. A 52, 3457 (1995)]. We also created a computer program which realizes both Barenco’s decomposition and the proposed optimization. Furthermore, our optimization can be applied to any quantum circuit containing generalized Toffoli gates, including basic quantum gate circuits.

scholarly journals Clifford recompilation for faster classical simulation of quantum circuits

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 170
Author(s):  
Hammam Qassim ◽  
Joel J. Wallman ◽  
Joseph Emerson
Keyword(s):  
Fault Tolerant ◽  
State Of The Art ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Monte Carlo Algorithm ◽  
Quantum Devices ◽  
Output State ◽  
Classical Simulation ◽  
Main Technique ◽  
Near Term

Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al, utilizes a randomized sparsification technique to approximate the output state of a quantum circuit by a stabilizer sum with a reduced number of terms. In this paper, we describe an improved Monte Carlo algorithm for performing randomized sparsification. This algorithm reduces the runtime of computing the approximate state by the factorℓ/m, whereℓandmare respectively the total and non-Clifford gate counts. The main technique is a circuit recompilation routine based on manipulating exponentiated Pauli operators. The recompilation routine also facilitates numerical search for Clifford decompositions of products of non-Clifford gates, which can further reduce the runtime in certain cases by reducing the 1-norm of the vector of expansion,‖a‖1. It may additionally lead to a framework for optimizing circuit implementations over a gate-set, reducing the overhead for state-injection in fault-tolerant implementations. We provide a concise exposition of randomized sparsification, and describe how to use it to estimate circuit amplitudes in a way which can be generalized to a broader class of gates and states. This latter method can be used to obtain additive error estimates of circuit probabilities with a faster runtime than the full techniques of Bravyi et al. Such estimates are useful for validating near-term quantum devices provided that the target probability is not exponentially small.

scholarly journals On the Design of Molecular Excitonic Circuits for Quantum Computing: The Universal Quantum Gates

2019 ◽  
Author(s):  
Maria Castellanos ◽  
Amro Dodin ◽  
Adam Willard
Keyword(s):  
Quantum Computing ◽  
Quantum Logic ◽  
Unitary Transformation ◽  
Design Space ◽  
Logic Gate ◽  
Quantum Gates ◽  
Dye Molecules ◽  
Quantum Logic Gate ◽  
Universal Quantum ◽  
Two Level Systems

This manuscript presents a theoretical strategy for encoding elementary quantum computing operations into the design of molecular excitonic circuits. Specifically, we show how the action of a unitary transformation of coupled two-level systems can be equivalently represented by the evolution of an exciton in a coupled network of dye molecules. We apply this strategy to identify the geometric parameters for circuits that perform universal quantum logic gate operations. We quantify the design space for these circuits and how their performance is affected by environmental noise.

scholarly journals EXACT NON-IDENTITY CHECK IS NQP-COMPLETE

2010 ◽  
Vol 08 (05) ◽  
pp. 807-819
Author(s):  
YU TANAKA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Decision Problem ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Unitary Operation ◽  
Classical Description ◽  
Quantum Polynomial ◽  
Equivalence Check

To understand quantum gate array complexity, we define a problem named exact non-identity check, which is a decision problem to determine whether a given classical description of a quantum circuit is strictly equivalent to the identity or not. We show that the computational complexity of this problem is non-deterministic quantum polynomial-time (NQP)-complete. As corollaries, it is derived that exact non-equivalence check of two given classical descriptions of quantum circuits is also NQP-complete and that minimizing the number of quantum gates for a given quantum circuit without changing the implemented unitary operation is NQP-hard.

scholarly journals Quantum advantage from energy measurements of many-body quantum systems

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 465
Author(s):  
Leonardo Novo ◽  
Juani Bermejo-Vega ◽  
Raúl García-Patrón
Keyword(s):  
High Resolution ◽  
Measurement Errors ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Polynomial Hierarchy ◽  
Many Body ◽  
Classical Simulation ◽  
Measurement Resolution ◽  
New Perspective ◽  
Sampling Problems

The problem of sampling outputs of quantum circuits has been proposed as a candidate for demonstrating a quantum computational advantage (sometimes referred to as quantum "supremacy"). In this work, we investigate whether quantum advantage demonstrations can be achieved for more physically-motivated sampling problems, related to measurements of physical observables. We focus on the problem of sampling the outcomes of an energy measurement, performed on a simple-to-prepare product quantum state – a problem we refer to as energy sampling. For different regimes of measurement resolution and measurement errors, we provide complexity theoretic arguments showing that the existence of efficient classical algorithms for energy sampling is unlikely. In particular, we describe a family of Hamiltonians with nearest-neighbour interactions on a 2D lattice that can be efficiently measured with high resolution using a quantum circuit of commuting gates (IQP circuit), whereas an efficient classical simulation of this process should be impossible. In this high resolution regime, which can only be achieved for Hamiltonians that can be exponentially fast-forwarded, it is possible to use current theoretical tools tying quantum advantage statements to a polynomial-hierarchy collapse whereas for lower resolution measurements such arguments fail. Nevertheless, we show that efficient classical algorithms for low-resolution energy sampling can still be ruled out if we assume that quantum computers are strictly more powerful than classical ones. We believe our work brings a new perspective to the problem of demonstrating quantum advantage and leads to interesting new questions in Hamiltonian complexity.

scholarly journals Householder methods for quantum circuit design

2016 ◽  
Vol 94 (2) ◽  
pp. 150-157 ◽  
Author(s):  
Jesús Urías ◽  
Diego A. Quiñones
Keyword(s):  
Tensor Product ◽  
Circuit Design ◽  
Quantum Circuit ◽  
Quantum Gates ◽  
Single Type ◽  
Unitary Operation ◽  
Gray Codes ◽  
Combined Procedure ◽  
Unitary Transformations ◽  
Assembly Procedure

Algorithms to resolve multiple-qubit unitary transformations into a sequence of simple operations on one-qubit subsystems are central to the methods of quantum-circuit simulators. We adapt Householder’s theorem to the tensor-product character of multi-qubit state vectors and translate it to a combinatorial procedure to assemble cascades of quantum gates that recreate any unitary operation U acting on n-qubit systems. U may be recreated by any cascade from a set of combinatorial options that, in number, are not lesser than super-factorial of 2n, [Formula: see text]. Cascades are assembled with one-qubit controlled-gates of a single type. We complement the assembly procedure with a new algorithm to generate Gray codes that reduce the combinatorial options to cascades with the least number of CNOT gates. The combined procedure —factorization, gate assembling, and Gray ordering — is illustrated on an array of three qubits.

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