scholarly journals Quantum Simulation Logic, Oracles, and the Quantum Advantage

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 800 ◽  
Author(s):  
Niklas Johansson ◽  
Jan-Åke Larsson
Keyword(s):  
Quantum Computer ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Quantum Simulation ◽  
Query Complexity ◽  
Integer Factorization ◽  
Simulation Framework ◽  
Shor's Algorithm ◽  
Quantum Query ◽  
Do So

Query complexity is a common tool for comparing quantum and classical computation, and it has produced many examples of how quantum algorithms differ from classical ones. Here we investigate in detail the role that oracles play for the advantage of quantum algorithms. We do so by using a simulation framework, Quantum Simulation Logic (QSL), to construct oracles and algorithms that solve some problems with the same success probability and number of queries as the quantum algorithms. The framework can be simulated using only classical resources at a constant overhead as compared to the quantum resources used in quantum computation. Our results clarify the assumptions made and the conditions needed when using quantum oracles. Using the same assumptions on oracles within the simulation framework we show that for some specific algorithms, such as the Deutsch-Jozsa and Simon’s algorithms, there simply is no advantage in terms of query complexity. This does not detract from the fact that quantum query complexity provides examples of how a quantum computer can be expected to behave, which in turn has proved useful for finding new quantum algorithms outside of the oracle paradigm, where the most prominent example is Shor’s algorithm for integer factorization.

scholarly journals Quantum query complexity of symmetric oracle problems

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 403
Author(s):  
Daniel Copeland ◽  
Jamie Pommersheim
Keyword(s):  
Polynomial Interpolation ◽  
Quantum Algorithms ◽  
Success Probability ◽  
Quantum Algorithm ◽  
Random Permutation ◽  
Query Complexity ◽  
Learning Problems ◽  
Unitary Matrices ◽  
Query Algorithm ◽  
Quantum Query

We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t-query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn. More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G, we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.

scholarly journals Exact quantum algorithms have advantage for almost all Boolean functions

2015 ◽  
pp. 435-452
Author(s):  
Andris Ambainis ◽  
Jozef Gruska ◽  
Shenggen Zheng
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Query Complexity ◽  
Exact Quantum ◽  
Quantum Query Complexity ◽  
Almost All ◽  
Quantum Query

It has been proved that almost all n-bit Boolean functions have exact classical query complexity n. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all n-bit Boolean functions can be computed by an exact quantum algorithm with less than n queries. More exactly, we prove that ANDn is the only n-bit Boolean function, up to isomorphism, that requires n queries.

scholarly journals Quantum algorithms for computing general discrete logarithms and orders with tradeoffs

2021 ◽  
Vol 15 (1) ◽  
pp. 359-407
Author(s):  
Martin Ekerå
Keyword(s):  
Quantum Algorithms ◽  
Probability Distributions ◽  
Success Probability ◽  
Discrete Logarithms ◽  
Group Order ◽  
Shor's Algorithm

Abstract We generalize our earlier works on computing short discrete logarithms with tradeoffs, and bridge them with Seifert's work on computing orders with tradeoffs, and with Shor's groundbreaking works on computing orders and general discrete logarithms. In particular, we enable tradeoffs when computing general discrete logarithms. Compared to Shor's algorithm, this yields a reduction by up to a factor of two in the number of group operations evaluated quantumly in each run, at the expense of having to perform multiple runs. Unlike Shor's algorithm, our algorithm does not require the group order to be known. It simultaneously computes both the order and the logarithm. We analyze the probability distributions induced by our algorithm, and by Shor's and Seifert's order-finding algorithms, describe how these algorithms may be simulated when the solution is known, and estimate the number of runs required for a given minimum success probability when making different tradeoffs.

scholarly journals Quantum algorithms for graph connectivity and formula evaluation

Quantum ◽  
2017 ◽  
Vol 1 ◽  
pp. 26
Author(s):  
Stacey Jeffery ◽  
Shelby Kimmel
Keyword(s):  
Quantum Algorithms ◽  
Graph Connectivity ◽  
Query Complexity ◽  
Boolean Formula ◽  
Quantum Query Complexity ◽  
Boolean Formulas ◽  
Speed Up ◽  
The Right ◽  
Quantum Query ◽  
Better Than

We give a new upper bound on the quantum query complexity of decidingst-connectivity on certain classes of planar graphs, and show the bound is sometimes exponentially better than previous results. We then show Boolean formula evaluation reduces to deciding connectivity on just such a class of graphs. Applying the algorithm forst-connectivity to Boolean formula evaluation problems, we match theO(N)bound on the quantum query complexity of evaluating formulas onNvariables, give a quadratic speed-up over the classical query complexity of a certain class of promise Boolean formulas, and show this approach can yield superpolynomial quantum/classical separations. These results indicate that thisst-connectivity-based approach may be the "right" way of looking at quantum algorithms for formula evaluation.

scholarly journals Qibo: a framework for quantum simulation with hardware acceleration

2021 ◽  
Author(s):  
Stavros Efthymiou ◽  
Sergi Ramos-Calderer ◽  
Carlos Bravo-Prieto ◽  
Adriian Perez-Salinas ◽  
Diego García-Martín ◽  
...  
Keyword(s):  
Open Source Software ◽  
Quantum Algorithms ◽  
Hardware Acceleration ◽  
Quantum Simulation ◽  
Quantum Circuits ◽  
Simulation Framework ◽  
Fast Evaluation ◽  
Simulation Performance ◽  
Recent Developments ◽  
User Friendly

Abstract We present Qibo, a new open-source software for fast evaluation of quantum circuits and adiabatic evolution which takes full advantage of hardware accelerators. The growing interest in quantum computing and the recent developments of quantum hardware devices motivates the development of new advanced computational tools focused on performance and usage simplicity. In this work we introduce a new quantum simulation framework that enables developers to delegate all complicated aspects of hardware or platform implementation to the library so they can focus on the problem and quantum algorithms at hand. This software is designed from scratch with simulation performance, code simplicity and user friendly interface as target goals. It takes advantage of hardware acceleration such as multi-threading CPU, single GPU and multi-GPU devices.

Introduction to Quantum Computing

2020 ◽  
pp. 259-264
Author(s):  
Andreas Bolfing
Keyword(s):  
Quantum Computing ◽  
Quantum Theory ◽  
Large Scale ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Public Key ◽  
Public Key Cryptosystems ◽  
Shor's Algorithm ◽  
Mathematical Problems ◽  
The Impact

This chapter gives a brief introduction to quantum computing, which is the discipline of studying algorithms based on the principles of quantum theory. It outlines the two fundamental quantum algorithms, which are known as Grover’s and Shor’s algorithm, which are able to solve number-theoretical problems that are intractable for present conventional computers. Thus, this chapter also shows the impact of these quantum algorithms on present cryptography under the assumption of the existence of a large-scale quantum computer, concluding that quantum computing poses a serious threat to public-key cryptosystems, because their underlying mathematical problems can be solved efficiently by using Shor’s algorithm.

scholarly journals Entanglement and its role in Shor's algorithm

2006 ◽  
Vol 6 (7) ◽  
pp. 630-640
Author(s):  
V.M. Kendon ◽  
W.J. Munro
Keyword(s):  
Numerical Simulation ◽  
Fourier Transform ◽  
Quantum Information ◽  
Quantum Information Processing ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Fourier Transform ◽  
Shor's Algorithm ◽  
Critical Resource ◽  
Factoring Algorithm

Entanglement has been termed a critical resource for quantum information processing and is thought to be the reason that certain quantum algorithms, such as Shor's factoring algorithm, can achieve exponentially better performance than their classical counterparts. The nature of this resource is still not fully understood: here we use numerical simulation to investigate how entanglement between register qubits varies as Shor's algorithm is run on a quantum computer. The shifting patterns in the entanglement are found to relate to the choice of basis for the quantum Fourier transform.

scholarly journals Quantum algorithms and approximating polynomials for composed functions with shared inputs

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 543
Author(s):  
Mark Bun ◽  
Robin Kothari ◽  
Justin Thaler
Keyword(s):  
Quantum Algorithms ◽  
Time Algorithm ◽  
Linear Size ◽  
Query Complexity ◽  
Inner Product ◽  
Worst Case ◽  
Subexponential Time ◽  
Quantum Query Complexity ◽  
Composed Functions ◽  
Quantum Query

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m-bit Boolean function and consider an n-bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f), we give an algorithm for evaluating F using O~(Q(f)⋅n) quantum queries. This improves on the bound of O(Q(f)⋅n) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f.By recursively applying our composition theorems, we obtain a nearly optimal O~(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0 circuits.As an additional consequence, we show that AC0∘⊕ circuits of depth d+1 require size Ω~(n1/(1−2−d))≥ω(n1+2−d) to compute the Inner Product function even on average. The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).

scholarly journals Practical Quantum Computing

2021 ◽  
Vol 2 (1) ◽  
pp. 1-35
Author(s):  
Adrien Suau ◽  
Gabriel Staffelbach ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Equation Solving ◽  
Small Quantum ◽  
Quantum Wave ◽  
Variational Algorithms ◽  
The One

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.

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