scholarly journals Quantum addition circuits and unbounded fan-out

2010 ◽  
Vol 10 (9&10) ◽  
pp. 872-890 ◽  
Author(s):  
Yasuhiro Takahashi ◽  
Seiichiro Tani ◽  
Noboru Kunihiro
Keyword(s):  
Positive Constant ◽  
Discrete Logarithm ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Small Positive Constant ◽  
Exact Size

We first show how to construct an $O(n)$-depth $O(n)$-size quantum circuit for addition of two $n$-bit binary numbers with no ancillary qubits. The exact size is $7n-6$, which is smaller than that of any other quantum circuit ever constructed for addition with no ancillary qubits. Using the circuit, we then propose a method for constructing an $O(d(n))$-depth $O(n)$-size quantum circuit for addition with $O(n/d(n))$ ancillary qubits for any $d(n) = \Omega(\log n)$. If we are allowed to use unbounded fan-out gates with length $O(n^{\varepsilon})$ for an arbitrary small positive constant $\varepsilon$, we can modify the method and construct an $O(e(n))$-depth $O(n)$-size circuit with $o(n)$ ancillary qubits for any $e(n) = \Omega(\log^* n)$. In particular, these methods yield efficient circuits with depth $O(\log n)$ and with depth $O(\log^* n)$, respectively. We apply our circuits to constructing efficient quantum circuits for Shor's discrete logarithm algorithm.

scholarly journals A quantum circuit to find discrete logarithms on ordinary binary elliptic curves in depth O(log^2n)

2014 ◽  
Vol 14 (9&10) ◽  
pp. 888-900
Author(s):  
Martin Rotteler ◽  
Rainer Steinwandt
Keyword(s):  
Elliptic Curves ◽  
Discrete Logarithm ◽  
Quantum Circuit ◽  
Discrete Logarithm Problem ◽  
Quantum Circuits ◽  
Discrete Logarithms ◽  
Shor's Algorithm

Improving over an earlier construction by Kaye and Zalka \cite{KaZa04}, in \cite{MMCP09b} Maslov et al. describe an implementation of Shor's algorithm, which can solve the discrete logarithm problem on ordinary binary elliptic curves in quadratic depth $\bigO(n^2)$. In this paper we show that discrete logarithms on such curves can be found with a quantum circuit of depth $\bigO(\log^2n)$. As technical tools we introduce quantum circuits for ${\mathbb F}_{2^n}$-multiplication in depth $\bigO(\log n)$ and for ${\mathbb F}_{2^n}$-inversion in depth $\bigO(\log^2 n)$.

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.

Uniform asymptotic expansions for oblate spheroidal functions I: positive separation parameter λ

1992 ◽  
Vol 121 (3-4) ◽  
pp. 303-320 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Wave Equation ◽  
Half Plane ◽  
Positive Constant ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Small Positive Constant ◽  
Separation Parameter ◽  
Oblate Spheroidal ◽  
Uniform Asymptotic Expansions

SynopsisUniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.

scholarly journals Spacetime as a quantum circuit

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
A. Ramesh Chandra ◽  
Jan de Boer ◽  
Mario Flory ◽  
Michal P. Heller ◽  
Sergio Hörtner ◽  
...  
Keyword(s):  
Path Integral ◽  
Constant Scalar Curvature ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Special Role ◽  
Gravitational Action ◽  
Volume Conjecture ◽  
Interesting Connection ◽  
Kinematic Space ◽  
Boundary States

Abstract We propose that finite cutoff regions of holographic spacetimes represent quantum circuits that map between boundary states at different times and Wilsonian cutoffs, and that the complexity of those quantum circuits is given by the gravitational action. The optimal circuit minimizes the gravitational action. This is a generalization of both the “complexity equals volume” conjecture to unoptimized circuits, and path integral optimization to finite cutoffs. Using tools from holographic $$ T\overline{T} $$ T T ¯ , we find that surfaces of constant scalar curvature play a special role in optimizing quantum circuits. We also find an interesting connection of our proposal to kinematic space, and discuss possible circuit representations and gate counting interpretations of the gravitational action.

Quantum generative adversarial networks for learning and loading quantum image in noisy environment

2021 ◽  
pp. 2150360
Author(s):  
Wanghao Ren ◽  
Zhiming Li ◽  
Yiming Huang ◽  
Runqiu Guo ◽  
Lansheng Feng ◽  
...  
Keyword(s):  
Image Processing ◽  
Quantum Circuit ◽  
Quantum Image Processing ◽  
Quantum Circuits ◽  
Generative Adversarial Networks ◽  
Channel Noise ◽  
Quantum Image ◽  
Adversarial Networks ◽  
Potential Applications ◽  
Classical Image

Quantum machine learning is expected to be one of the potential applications that can be realized in the near future. Finding potential applications for it has become one of the hot topics in the quantum computing community. With the increase of digital image processing, researchers try to use quantum image processing instead of classical image processing to improve the ability of image processing. Inspired by previous studies on the adversarial quantum circuit learning, we introduce a quantum generative adversarial framework for loading and learning a quantum image. In this paper, we extend quantum generative adversarial networks to the quantum image processing field and show how to learning and loading an classical image using quantum circuits. By reducing quantum gates without gradient changes, we reduced the number of basic quantum building block from 15 to 13. Our framework effectively generates pure state subject to bit flip, bit phase flip, phase flip, and depolarizing channel noise. We numerically simulate the loading and learning of classical images on the MINST database and CIFAR-10 database. In the quantum image processing field, our framework can be used to learn a quantum image as a subroutine of other quantum circuits. Through numerical simulation, our method can still quickly converge under the influence of a variety of noises.

scholarly journals Error mitigation with Clifford quantum-circuit data

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 592
Author(s):  
Piotr Czarnik ◽  
Andrew Arrasmith ◽  
Patrick J. Coles ◽  
Lukasz Cincio
Keyword(s):  
State Energy ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Training Data ◽  
Quantum Circuits ◽  
Accurate Estimation ◽  
Error Mitigation ◽  
Order Of Magnitude ◽  
Quantum Observables ◽  
Near Term

Achieving near-term quantum advantage will require accurate estimation of quantum observables despite significant hardware noise. For this purpose, we propose a novel, scalable error-mitigation method that applies to gate-based quantum computers. The method generates training data {Xinoisy,Xiexact} via quantum circuits composed largely of Clifford gates, which can be efficiently simulated classically, where Xinoisy and Xiexact are noisy and noiseless observables respectively. Fitting a linear ansatz to this data then allows for the prediction of noise-free observables for arbitrary circuits. We analyze the performance of our method versus the number of qubits, circuit depth, and number of non-Clifford gates. We obtain an order-of-magnitude error reduction for a ground-state energy problem on 16 qubits in an IBMQ quantum computer and on a 64-qubit noisy simulator.

scholarly journals An Innovative Genetic Algorithm for the Quantum Circuit Compilation Problem

2019 ◽  
Vol 33 ◽  
pp. 7707-7714
Author(s):  
Riccardo Rasconi ◽  
Angelo Oddi
Keyword(s):  
Genetic Algorithm ◽  
Nearest Neighbor ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Heuristic Function ◽  
Circuit Realization ◽  
Benchmark Instances ◽  
Technological Limitations ◽  
Selection Of

Quantum Computing represents the next big step towards speed boost in computation, which promises major breakthroughs in several disciplines including Artificial Intelligence. This paper investigates the performance of a genetic algorithm to optimize the realization (compilation) of nearest-neighbor compliant quantum circuits. Currrent technological limitations (e.g., decoherence effect) impose that the overall duration (makespan) of the quantum circuit realization be minimized, and therefore the makespanminimization problem of compiling quantum algorithms on present or future quantum machines is dragging increasing attention in the AI community. In our genetic algorithm, a solution is built utilizing a novel chromosome encoding where each gene controls the iterative selection of a quantum gate to be inserted in the solution, over a lexicographic double-key ranking returned by a heuristic function recently published in the literature.Our algorithm has been tested on a set of quantum circuit benchmark instances of increasing sizes available from the recent literature. We demonstrate that our genetic approach obtains very encouraging results that outperform the solutions obtained in previous research against the same benchmark, succeeding in significantly improving the makespan values for a great number of instances.

Quantum circuits for calculating the squared sum of the inner product of quantum states and its application

2019 ◽  
Vol 17 (05) ◽  
pp. 1950043
Author(s):  
Panchi Li ◽  
Jiahui Guo ◽  
Bing Wang ◽  
Mengqi Hao
Keyword(s):  
Quantum Circuit ◽  
Quantum States ◽  
Experimental Results ◽  
Quantum Circuits ◽  
Inner Product ◽  
Superposition State ◽  
Initial State ◽  
Control Rules ◽  
Control Qubit ◽  
Classical Computer

In this paper, we propose a quantum circuit for calculating the squared sum of the inner product of quantum states. The circuit is designed by the multi-qubits controlled-swapping gates, in which the initial state of each control qubit is [Formula: see text] and they are in the equilibrium superposition state after passing through some Hadamard gates. Then, according to the control rules, each basis state in the superposition state controls the corresponding quantum states pair to swap. Finally, the Hadamard gates are applied to the control qubits again, and the squared sum of the inner product of many pairs of quantum states can be obtained simultaneously by measuring only one control qubit. We investigate the application of this method in quantum images matching on a classical computer, and the experimental results verify the correctness of the proposed method.

scholarly journals Revisiting the simulation of quantum Turing machines by quantum circuits

2019 ◽  
Vol 475 (2226) ◽  
pp. 20180767 ◽  
Author(s):  
Abel Molina ◽  
John Watrous
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Computational Models ◽  
Circuit Complexity ◽  
Simulation Method ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Turing Machines ◽  
Generated Family ◽  
Quantum Turing Machine

Yao's 1995 publication ‘Quantum circuit complexity’ in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science , pp. 352–361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t , and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t , rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77 , 372–378. ( doi:10.1016/j.jcss.2010.05.004 ), that allows for the localization of causal unitary evolutions.

scholarly journals Optimal ILP-Based Approach for Gate Location Assignment and Scheduling in Quantum Circuits

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Naser Mohammadzadeh ◽  
Tayebeh Bahreini ◽  
Hossein Badri
Keyword(s):  
Circuit Design ◽  
Design Methodology ◽  
Ion Trap ◽  
Quantum Circuit ◽  
Physical Design ◽  
Quantum Circuits ◽  
Mixed Integer ◽  
Routing And Scheduling ◽  
Design And Synthesis ◽  
Proposed Model

Physical design and synthesis are two key processes of quantum circuit design methodology. The physical design process itself decomposes into scheduling, mapping, routing, and placement. In this paper, a mathematical model is proposed for mapping, routing, and scheduling in ion-trap technology in order to minimize latency of the circuit. The proposed model which is a mixed integer linear programming (MILP) model gives the optimal locations for gates and the best sequence of operations in terms of latency. Experimental results show that our scheme outperforms the other schemes for the attempted benchmarks.

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