9 text pages, 3 figures in one additional ps file, manuscript of the presentation to be held at the SILFS Workshop on Logic and Quantum Computation, Cesena, Italy, February 16, 1999
A typical oracle problem is finding which software program is installed on a computer, by running the computer and testing its input-output behaviour. The program is randomly chosen from a set of programs known to the problem solver. As well known, some oracle problems are solved more efficiently by using quantum algorithms; this naturally implies changing the computer to quantum, while the choice of the software program remains sharp. In order to highlight the non-mechanistic origin of this higher efficiency, also the uncertainty about which program is installed must be represented in a quantum way.
Computational devices may be supplied with external sources of information (oracles). Quantum oracles may transmit phase information which is available to a quantum computer but not a classical computer. One consequence of this observation is that there is an oracle which is of no assistance to a classical computer but which allows a quantum computer to solve undecidable problems. Thus useful relativized separations between quantum and classical complexity classes must exclude the transmission of phase information from oracle to computer.
We describe a new implementation of the Bernstein-Vazirani algorithm which relies on the fact that the polarization states of classical light beams can be cloned. We explore the possibility of computing with waves and discuss a classical optical model capable of implementing any algorithm (on $n$ qubits) that does not involve entanglement. The Bernstein-Vazirani algorithm (with a suitably modified oracle), wherein a hidden $n$ bit vector is discovered by one oracle query as against $n$ oracle queries required classically, belongs to this category. In our scheme, the modified oracle is also cap…
Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schrödinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For…
Hamiltonian oracles are the continuum limit of the standard unitary quantum oracles. In this limit, the problem of finding the optimal query algorithm can be mapped into the problem of finding shortest paths on a manifold. The study of these shortest paths leads to lower bounds of the original unitary oracle problem. A number of example Hamiltonian oracles are studied in this paper, including oracle interrogation and the problem of computing the XOR of the hidden bits. Both of these problems are related to the study of geodesics on spheres with non-round metrics. For the case of two hidden bit…
Recently a great deal of attention has focused on quantum computation following a sequence of results suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of NP can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turing machine in…
We present a framework for efficiently solving Approximate Traveling Salesman Problem (Approximate TSP) for Quantum Computing Models. Existing representations of TSP introduce extra states which do not correspond to any permutation. We present an efficient and intuitive encoding for TSP in quantum computing paradigm. Using this representation and assuming a Gaussian distribution on tour-lengths, we give an algorithm to solve Approximate TSP (Euclidean) within BQP resource bounds. Generalizing this strategy for any distribution, we present an oracle based Quantum Algorithm to solve Approximate …
The main promise of quantum computing is to efficiently solve certain problems that are prohibitively expensive for a classical computer. Most problems with a proven quantum advantage involve the repeated use of a black box, or oracle, whose structure encodes the solution. One measure of the algorithmic performance is the query complexity, i.e., the scaling of the number of oracle calls needed to find the solution with a given probability. Few-qubit demonstrations of quantum algorithms, such as Deutsch-Jozsa and Grover, have been implemented across diverse physical systems such as nuclear magn…
In the standard oracle model, an oracle efficiently evaluates an unknown classical function independent of the quantum algorithm itself. Quantum algorithms have a complex interrelationship to their oracles; for example the possibility of quantum speedup is affected by the manner by which oracles are implemented. Therefore, it is physically meaningful to separate oracles from their quantum algorithms, and we introduce one such separation here. We define the Blind Oracular Quantum Computation (BOQC) scheme, in which the oracle is a distinct node in a quantum network. Our work augments the client…
A dynamical quantum model assigns an eigenstate to a specified observable even when no measurement is made, and gives a stochastic evolution rule for that eigenstate. Such a model yields a distribution over classical histories of a quantum state. We study what can be computed by sampling from that distribution, i.e., by examining an observer's entire history. We show that, relative to an oracle, one can solve problems in polynomial time that are intractable even for quantum computers; and can search an N-element list in order N^{1/3} steps (though not fewer).
Key Laboratory of System Software (Chinese Academy of Sciences), Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所系统软件重点实验室）；Department of Computer Science and Technology, Tsinghua University（清华大学计算机科学与技术系）；Centre for Quantum Software and Information, University of Technology Sydney（悉尼科技大学量子软件与信息中心）
Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper …
Selecting a set of basis states is a common task in quantum computing, in order to increase and/or evaluate their probabilities. This is similar to designing WHERE clauses in classical database queries. Even though one can find heuristic methods to achieve this, it is desirable to automate the process. A common, but inefficient automation approach is to use oracles with classical evaluation of all the states at circuit design time. In this paper, we present a novel, canonical way to produce a quantum oracle from an algebraic expression (in particular, an Ising model), that maps a set of select…
It is known that the group non-membership problem is in QMA relative to any group oracle and in ${\rm SPP}\cap{\rm BQP}$ relative to group oracles for solvable groups. We consider a modified version of the group non-membership problem where the order of the group is also given as an additional input. We show that the problem is in AWPP relative to any group oracle. To show the result, we use the idea of the postselected quantum computing.
Fraunhofer FIRST develops a computing service and collaborative workspace providing a convenient tool for simulation and investigation of quantum algorithms. To broaden the twenty qubit limit of workstation-based simulations to the next qubit decade we provide a dedicated high memorized Linux cluster with fast Myrinet interconnection network together with a adapted parallel simulator engine. This simulation service supplemented by a collaborative workspace is usable everywhere via web interface and integrates both hardware and software as collaboration and investigation platform for the quantu…
Our problem is to evaluate a multi-valued Boolean function $F$ through oracle calls. If $F$ is one-to-one and the size of its domain and range is the same, then our problem can be formulated as follows: Given an oracle $f(a,x): \{0,1\}^n\times\{0,1\}^n \to \{0,1\}$ and a fixed (but hidden) value $a_0$, we wish to obtain the value of $a_0$ by querying the oracle $f(a_0,x)$. Our goal is to minimize the number of such oracle calls (the query complexity) using a quantum mechanism. Two popular oracles are the EQ-oracle defined as $f(a,x)=1$ iff $x=a$ and the IP-oracle defined as $f(a,x)= a\cdot x \…
The main idea in the original Grover search (Phys. Rev. Lett. 79, 325 (1997)) is to single out a target state containing the solution to a search problem by amplifying the amplitude of the state, following the Oracle's job, i.e., a black box giving us information about the target state. We design quantum circuits to accomplish a complete Grover search involving both the Oracle's job and the amplification of the target state, which are employed to solve Satisfiability (SAT) problems. We explore how to carry out the quantum circuits by currently available ion-trap quantum computing technology.
The steadily growing research interest in quantum computing - together with the accompanying technological advances in the realization of quantum hardware - fuels the development of meaningful real-world applications, as well as implementations for well-known quantum algorithms. One of the most prominent examples till today is Grover's algorithm, which can be used for efficient search in unstructured databases. Quantum oracles that are frequently masked as black boxes play an important role in Grover's algorithm. Hence, the automatic generation of oracles is of paramount importance. Moreover, …
Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a probability as close to 1 as desired. The algorithm requires $O((\log n )^3n^3)$ quantum operations, or $O(\log\log n (\log n)^4 n^2)$ operations using fast multiplication. Verification, quantum optimizations and applications to RSA and primality tests are also discussed.
The aim of this work is to show a brand-new way of making deterministic Quantum Computing (short QC), in the sense of Theory of Calculability, by meaning of unitary evolution. We start from the original Shor's Algorithm to explain how the newest one works, at least compared to theory. We will give a new conceptual foundation of QC, resulting from a set of conventional and well known results of Calculability and Quantum Mechanics. In the practice, if that can be used in its general sense, we will show an inaccessible relativized process which let us able to obtain same results with the same out…
This paper shows that, if we could examine the entire history of a hidden variable, then we could efficiently solve problems that are believed to be intractable even for quantum computers. In particular, under any hidden-variable theory satisfying a reasonable axiom called "indifference to the identity," we could solve the Graph Isomorphism and Approximate Shortest Vector problems in polynomial time, as well as an oracle problem that is known to require quantum exponential time. We could also search an N-item database using O(N^{1/3}) queries, as opposed to O(N^{1/2}) queries with Grover's sea…
The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision …
We present a constructive method to create quantum circuits that implement oracles $|x\rangle|y\rangle|0\rangle^k \mapsto |x\rangle|y \oplus f(x)\rangle|0\rangle^k$ for $n$-variable Boolean functions $f$ with low $T$-count. In our method $f$ is given as a 2-regular Boolean logic network over the gate basis $\{\land, \oplus, 1\}$. Our construction leads to circuits with a $T$-count that is at most four times the number of AND nodes in the network. In addition, we propose a SAT-based method that allows us to trade qubits for $T$ gates, and explore the space/complexity trade-off of quantum circui…
The recent debate on hypercomputation has arisen new questions both on the computational abilities of quantum systems and the Church-Turing Thesis role in Physics. We propose here the idea of "effective physical process" as the essentially physical notion of computation. By using the Bohm and Hiley active information concept we analyze the differences between the standard form (quantum gates) and the non-standard one (adiabatic and morphogenetic) of Quantum Computing, and we point out how its Super-Turing potentialities derive from an incomputable information source in accordance with Bell's c…
In this article we show how the structure of Coxeter groups are present in gate sets of reversible and quantum computing. These groups have efficient word problems which means that circuits built from these gates have potential to be shortened efficiently. This is especially useful in the case of quantum computing when one does not have the timescale to perform a long series of gates and so one must and a gate scheduling that minimizes circuit depth. As the main example we consider the oracle for 3SAT.
Fault-Class-Matched Test Oracles for Output-Invisible Quantum Transpiler Regressions
City University of Science and Information Technology (CUSIT)（城市科学与技术大学）；National University of Computer and Emerging Sciences (FAST-NUCES)（国家计算机与新兴科学大学）
We question whether the measurement based quantum computing algorithm is in fact Grover's algorithm or simply a similar oracular search method. The two algorithms share several qualitative features especially in the case of the trivial 4 element search, which is the largest size photonic search algorithm that has been experimentally implemented to date. This has led some to refer to both substantiations as Grover's algorithm. We compare multiple features of the two algorithms including the behavior of the oracle tags and the entanglement dynamics, both qualitatively and quantitatively. We find…
It is generally believed that entanglement is essential for quantum computing. We present here a few simple examples in which quantum computing without entanglement is better than anything classically achievable, in terms of the reliability of the outcome after a xed number of oracle calls. Using a separable (that is, unentangled) n-qubit state, we show that the Deutsch-Jozsa problem and the Simon problem can be solved more reliably by a quantum computer than by the best possible classical algorithm, even probabilistic. We conclude that: (a) entanglement is not essential for quantum computing;…
