A standard quantum oracle $S_f$ for a general function $f: Z_N \to Z_N $ is defined to act on two input states and return two outputs, with inputs $\ket{i}$ and $\ket{j}$ ($i,j \in Z_N $) returning outputs $\ket{i}$ and $\ket{j \oplus f(i)}$. However, if $f$ is known to be a one-to-one function, a simpler oracle, $M_f$, which returns $\ket{f(i)}$ given $\ket{i}$, can also be defined. We consider the relative strengths of these oracles. We define a simple promise problem which minimal quantum oracles can solve exponentially faster than classical oracles, via an algorithm which cannot be naively adapted to standard quantum oracles. We show that $S_f$ can be constructed by invoking $M_f$ and $(M_f)^{-1}$ once each, while $Θ(\sqrt{N})$ invocations of $S_f$ and/or $(S_f)^{-1}$ are required to construct $M_f$.
The oracle model of computation is believed to allow a rigorous proof of quantum over classical computational superiority. Since quantum and classical oracles are essentially different, a correspondence principle is commonly implicitly used as a platform for comparison of oracle complexity. Here, we question the grounds on which this correspondence is based. Obviously, results on quantum speed-up depend on the chosen correspondence. So, we introduce the notion of genuine quantum speed-up which can serve as a tool for reliable comparison of quantum vs classical complexity, independently of the …
It is an established fact that for many of the interesting problems quantum algorithms based on queries of the standard oracle bring no significant improvement in comparison to known classical algorithms. It is conceivable that there are other oracles of algorithmic importance acting in a less intuitive fashion to which such limitations do not apply. Thus motivated this article suggests a broader understanding towards what a general quantum oracle is. We propose a general definition of a quantum oracle and give a classification of quantum oracles based on the behavior of the eigenvalues and ei…
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 consider Grover's unstructured search problem in the setting where each oracle call has some small probability of failing. We show that no quantum speed-up is possible in this case.
Quantum oracles play key roles in the studies of quantum computation and quantum information. But implementing quantum oracles efficiently with universal quantum gates is a hard work. Motivated by genetic programming, this paper proposes a novel approach to evolve quantum oracles with a hybrid quantum-inspired evolutionary algorithm. The approach codes quantum circuits with numerical values and combines the cost and correctness of quantum circuits into the fitness function. To speed up the calculation of matrix multiplication in the evaluation of individuals, a fast algorithm of matrix multipl…
Automatic oracle generation techniques can find optimized quantum circuits for classical components in quantum algorithms. However, most implementations of oracle generation techniques require that the classical component is expressed in terms of a conventional logic representation such as logic networks, truth tables, or decision diagrams. We implemented LLVM passes that can automatically generate QIR functions representing classical Q# functions into QIR code implementing such functions quantumly. We are using state-of-the-art logic optimization and oracle generation techniques based on XOR-…
This paper shows that, in measurement-based quantum computation, it is possible to write any quantum oracle implementing a classical function in constant depth. The result is shown through the equivalence between MBQC and the circuit model where arbitrary rotations along $Z$ axis and unbounded fan-outs are elementary operations. A corollary of this result is that disjunction can be implemented exactly in constant-depth, answering an open question of Høyer and Špalek.
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 r…
Grover's algorithm has achieved great success. But quantum search algorithms still are not complete algorithms because of Grover's Oracle. We concerned on this problem and present a new quantum search algorithm in adiabatic model without Oracle. We analyze the general difficulties in quantum search algorithms and show how to solve them in the present algorithm. As well this algorithm could deal with both single-solution and multi-solution searches without modification. We also implement this algorithm on NMR quantum computer. It is the first experiment which perform a real quantum database sea…
This paper has been withdrawn by the authors due to a problem with *efficiently* predicting the large fourier coefficients. It is being reworked and will be resubmitted in the near future.
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 quant…
Using Grover algorithm, this work investigates a technique for encoding search phrases used in wildcard searches. The technique involves creating a phase Oracle that loads data into a quantum circuit together with the search terms that have been encoded. The method entails constructing a phase Oracle programmatically using encoded input information and encoded search terms. By combining Grover diffusion operator with the phase Oracle, Hadamard gates, and zero-initialized three-qubit states, a complete quantum circuit is created. Trapped ion and superconducting qubit quantum computers, which we…
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…
This paper considers the quantum query complexity of {\it $\eps$-biased oracles} that return the correct value with probability only $1/2 + \eps$. In particular, we show a quantum algorithm to compute $N$-bit OR functions with $O(\sqrt{N}/{\eps})$ queries to $\eps$-biased oracles. This improves the known upper bound of $O(\sqrt{N}/{\eps}^2)$ and matches the known lower bound; we answer the conjecture raised by the paper by Iwama et al. affirmatively. We also show a quantum algorithm to cope with the situation in which we have no knowledge about the value of $\eps$. This contrasts with the corr…
How to implement quantum oracle with limited resources raises concerns these days. We design two ancilla-adjustable and efficient algorithms to synthesize SAT-oracle, the key component in solving SAT problems. The previous work takes 2m-1 ancillary qubits and O(m) elementary gates to synthesize an m clauses oracle. The first algorithm reduces the number of ancillary qubits to 2\sqrt{m}, with at most an eightfold increase in circuit size. The number of ancillary qubits can be further reduced to 3 with a quadratic increase in circuit size. The second algorithm aims to reduce the circuit depth. B…
We identify a key difference between quantum search by discrete- and continuous-time quantum walks: a discrete-time walk typically performs one walk step per oracle query, whereas a continuous-time walk can effectively perform multiple walk steps per query while only counting query time. As a result, we show that continuous-time quantum walks can outperform their discrete-time counterparts, even though both achieve quadratic speedups over their corresponding classical random walks. To provide greater equity, we allow the discrete-time quantum walk to also take multiple walk steps per oracle qu…
In this paper, we consider a quantum algorithm for solving the following problem: ``Suppose $f$ is a function given as a black box (that is also called an oracle) and $f$ is invariant under some AND-mask. Examine a property of $f$ by querying the oracle.'' We compare the efficiency of our quantum algorithm with that of classical algorithms by evaluating the expected number of queries for each algorithm. We show that our quantum algorithm is more efficient than any classical algorithm in some cases. However, our quantum algorithm does not exhibit an exponential speedup in the size of an input, …
Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle problems, including the GROUP MULTIPLICATION problem, in which the product of an ordered $n$-tuple of group elements is to be determined by querying elements of the tuple. For any finite group $G$, we give an algorithm to find the product of two elements of $G$ with a single quantum query with probability $2/|G|$. This generalizes Deutsch's Algorithm from $Z_…
This paper describes a novel approach to solving unstructured search problems using a classical, signal-based emulation of a quantum computer. The classical nature of the representation allows one to perform subspace projections in addition to the usual unitary gate operations. Although bandwidth requirements will limit the scale of problems that can be solved by this method, it can nevertheless provide a significant computational advantage for problems of limited size. In particular, we find that, for the same number of noisy oracle calls, the proposed subspace projection method provides a hi…
This paper has been withdrawn by Christino Tamon and Tomoyuki Yamakami because of errors of the main theorems. An erratum appeared in the Proceedings of the UMC 2000 Conference published by Springer.
We study a simple-harmonic-oscillator quantum computer solving oracle decision problems. We show that such computers can perform better by using nonorthogonal Gaussian wave functions rather than orthogonal top-hat wave functions as input to the information encoding process. Using the Deutsch-Jozsa problem as an example, we demonstrate that Gaussian modulation with optimized width parameter results in a lower error rate than for the top-hat encoding. We conclude that Gaussian modulation can allow for an improved trade-off between encoding, processing and measurement of the information.
We propose a quantum string comparison method whose main building blocks are a specially designed oracle construction followed by Grover's search algorithm. The purpose of the oracle is to compare all alphabets of the string in parallel. This requires a unique input state preparation, which when combined with some ancillas will result in a deterministic binary success and failure compare outcome.
The oracle identification problem (OIP) was introduced by Ambainis et al. \cite{AIKMRY04}. It is given as a set $S$ of $M$ oracles and a blackbox oracle $f$. Our task is to figure out which oracle in $S$ is equal to the blackbox $f$ by making queries to $f$. OIP includes several problems such as the Grover Search as special cases. In this paper, we improve the algorithms in \cite{AIKMRY04} by providing a mostly optimal upper bound of query complexity for this problem: ($i$) For any oracle set $S$ such that $|S| \le 2^{N^d}$ ($d < 1$), we design an algorithm whose query complexity is $O(\sqrt{N…
We present a systematic construction of quantum circuits implementing Grover's database search algorithm for arbitrary number of targets. We introduce a new operator which flips the sign of the targets and evaluate its circuit complexity. We find the condition under which the circuit complexity of the database search algorithm based on this operator is less than that of the conventional one.
This paper positively solves the quantum subroutine problem for fully quantum oracles. The quantum subroutine problem asks whether a quantum computer with an efficiently computable oracle can be efficiently simulated by a non-oracle quantum computer. We extends the earlier results obtained by Bennett, Bernstein, Brassard, and Vazirani, and by Aharonov, Kitaev, and Nisan to the case where the oracle evaluates a unitary operator and the computer is allowed to be in the superposition of a query state and a non-query state during computation. We also prove the robustness of {\bf EQP}, {\bf BQP}, a…
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 \…
Quantum information is well-known to achieve cryptographic feats that are unattainable using classical information alone. Here, we add to this repertoire by introducing a new cryptographic functionality called uncloneable encryption. This functionality allows the encryption of a classical message such that two collaborating but isolated adversaries are prevented from simultaneously recovering the message, even when the encryption key is revealed. Clearly, such functionality is unattainable using classical information alone. We formally define uncloneable encryption, and show how to achieve it …
The ZX-calculus was introduced as a graphical language able to represent specific quantum primitives in an intuitive way. The recent completeness results have shown the theoretical possibility of a purely graphical description of quantum processes. However, in practice, such approaches are limited by the intrinsic low level nature of ZX calculus. The scalable notations have been proposed as an attempt to recover an higher level point of view while maintaining the topological rewriting rules of a graphical language. We demonstrate that the scalable ZX-calculus provides a formal, intuitive, and …
We establish a framework for oracle identification problems in the continuous variable setting, where the stated problem necessarily is the same as in the discrete variable case, and continuous variables are manifested through a continuous representation in an infinite-dimensional Hilbert space. We apply this formalism to the Deutsch-Jozsa problem and show that, due to an uncertainty relation between the continuous representation and its Fourier-transform dual representation, the corresponding Deutsch-Jozsa algorithm is probabilistic hence forbids an exponential speed-up, contrary to a previou…
The interest in post-quantum cryptography - classical systems that remain secure in the presence of a quantum adversary - has generated elegant proposals for new cryptosystems. Some of these systems are set in the random oracle model and are proven secure relative to adversaries that have classical access to the random oracle. We argue that to prove post-quantum security one needs to prove security in the quantum-accessible random oracle model where the adversary can query the random oracle with quantum states. We begin by separating the classical and quantum-accessible random oracle models by…
Many quantum programs require circuits for addition, subtraction and logical operations. These circuits may be packaged within routines known as oracles. However, oracles can be tedious to code with current frameworks. To solve this problem the author developed Higher-Level Oracle Description Language (HODL) -- a C-style programming language for use on quantum computers -- to ease the creation of such circuits. The compiler translates high-level code written in HODL and converts it into OpenQASM, a gate-based quantum assembly language that runs on IBM Quantum Systems and compatible simulators.
We propose a genetic-algorithm-based method to find the unitary transformations for any desired quantum computation. We formulate a simple genetic algorithm by introducing the "genetic parameter vector" of the unitary transformations to be found. In the genetic algorithm process, all components of the genetic parameter vectors are supposed to evolve to the solution parameters of the unitary transformations. We apply our method to find the optimal unitary transformations and to generalize the corresponding quantum algorithms for a realistic problem, the one-bit oracle decision problem, or the o…
An algorithm for structured database searching is presented and used to solve the set partition problem. O(n) oracle calls are required in order to obtain a solution, but the probability that this solution is optimal decreases exponentially with problem size. Each oracle call is followed by a measurement, implying that it is necessary to maintain quantum coherence for only one oracle call at a time.
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 …
The evolution of quantum hardware is highlighting the need for advances in quantum software engineering that help developers create quantum software with good quality attributes. Specifically, reusability has been traditionally considered an important quality attribute in terms of efficiency of cost and effort. Increasing the reusability of quantum software will help developers create more complex solutions, by reusing simpler components, with better quality attributes, as long as the reused components have also these attributes. This work focuses on the reusability of oracles, a well-known pa…
Grover's search algorithm was originally proposed for circuit-based quantum computers. A crucial part of it is to query an oracle -- a black-box unitary operation. Generation of this oracle is formally beyond the original algorithm design. Here, we propose a realization of Grover's oracle for a large class of searching problems using a quantum annealing step. The time of this step grows only logarithmically with the size of the database. This suggests an efficient path to application of Grover's algorithm to practically important problems, such as finding the ground state of a Hamiltonian with…
Grover's quantum search algorithm provides a quadratic quantum advantage over classical algorithms across a broad class of unstructured search problems. The original protocol is probabilistic, returning the desired result with significant probability on each query, but in general, requiring several iterations of the algorithm. We present a modified version to return the correct result with certainty without having user control over the quantum search oracle. Our deterministic, two-parameter "D2p" protocol utilizes generalized phase rotations replacing the phase inversions after a standard orac…
A quantum oracle separation between QMA(2) and QMA
Simons Institute & UC Berkeley（西蒙斯研究所与加州大学伯克利分校）；IBM Research, Columbia University（IBM 研究部，哥伦比亚大学）；Ruhr-University Bochum（波鸿鲁尔大学）；University of California, Berkeley（加州大学伯克利分校）；MIT（麻省理工学院）；University of Washington（华盛顿大学）
Linear-depth quantum oracles for clique problems from edge colorings and graph states, with linear non-Clifford cost and a provably bounded-error k-clique search
School of Electrical and Computer Engineering, College of Engineering, University of Tehran（德黑兰大学工程学院电气与计算机工程学院）
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）
Quantum algorithms are typically understood in terms of the evolution of a multi-qubit quantum system under a prescribed sequence of unitary transformations. The input to the algorithm prescribes some of the unitary transformations in the sequence with others remaining fixed. For oracle query algorithms, the input determines the oracle unitary transformation. Such algorithms can be regarded as devices for discriminating amongst a set of unitary transformations. The question arises: "Given a set of known oracle unitary transformations, to what extent is it possible to discriminate amongst them?…
