An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung & Lai (STOC 2020) and Coudron & Menda (STOC 2020) showed that these models (with d=log^O(1) (n)) are strictly weaker than BQP (the class of problems solvable by poly-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. We show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d $\nsubseteq$ QC_d and QC_d $\nsubseteq$ CQ_d relative to an oracle. In other words, there exist problems that one model can solve but not the other and vice versa. We do this by considering new oracle problems that capture the distinctions between the two models and by introducing the notion of an intrinsically stochastic oracle, an oracle whose responses are inherently randomised, which is used for our second result. While we leave showing the second separation relative to a standard oracle as an open problem, we believe the notion of stochastic oracles could be of independent interest for studying complexity classes which have resisted separation in the standard oracle model. Our constructions also yield simpler oracle separations between the hybrid models and BQP, compared to earlier works.
After presentations of Raz and Tal's oracle separation of BQP and PH result, several people (e.g. Ryan O'Donnell, James Lee, Avishay Tal) suggested that the proof may be simplified by stochastic calculus. In this short note, we describe such a simplification.
A foundational question in quantum computational complexity asks how much more useful a quantum state can be in a given task than a comparable, classical string. Aaronson and Kuperberg showed such a separation in the presence of a quantum oracle, a black box unitary callable during quantum computation. Their quantum oracle responds to a random, marked, quantum state, which is intractable to specify classically. We constrain the marked state in ways that make it easy to specify classically while retaining separations in task complexity. Our method replaces query by state complexity. Furthermore…
We study the relationship between problems solvable by quantum algorithms in polynomial time and those for which zero-knowledge proofs exist. In prior work, Aaronson [arxiv:quant-ph/0111102] showed an oracle separation between BQP and SZK, i.e. an oracle $A$ such that $\mathrm{SZK}^A \not\subseteq \mathrm{BQP}^A$. In this paper we give a simple extension of Aaronson's result to non-interactive zero-knowledge proofs with perfect security. This class, NIPZK, is the most restrictive zero-knowledge class. We show that even for this class we can construct an $A$ with $\mathrm{NIPZK}^A \not\subseteq…
In 2009, using the $\textsf{Fourier Checking}$ problem, Aaronson claimed to construct the relativized worlds such that $\textsf{BQP} \not\subset \mathsf{BPP_{path}}$ and $\textsf{BQP} \not\subset \textsf{SZK}$. However, there are subtle errors in the original proof. In this paper, we point out the issues, and rescue these two separations by using more sophisticated constructions. Meanwhile, we take the opportunity to study the complexity classes $\mathsf{BPP_{path}}$ and $\textsf{SZK}$. We give general ways to construct functions which are hard for $\textsf{SZK}$ and $\mathsf{BPP_{path}}$ (in …
Theoretical computer scientists have been debating the role of oracles since the 1970's. This paper illustrates both that oracles can give us nontrivial insights about the barrier problems in circuit complexity, and that they need not prevent us from trying to solve those problems. First, we give an oracle relative to which PP has linear-sized circuits, by proving a new lower bound for perceptrons and low- degree threshold polynomials. This oracle settles a longstanding open question, and generalizes earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More importantly, it impl…
This paper investigates the power of quantum statistical zero knowledge interactive proof systems in the relativized setting. We prove the existence of an oracle relative to which quantum statistical zero-knowledge does not contain UP intersect coUP, and we prove that quantum statistical zero knowledge does not contain UP relative to a random oracle with probability 1. Our proofs of these statements rely on a bound on output state discrimination for relativized quantum circuits based on the quantum adversary method of Ambainis, following a technique similar to one used by Ben-David and Kothari…
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…
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…
Non-Standard Oracles for Bounded-Error Complexity Classes
Institute for Quantum Computing（量子计算研究所）；University of Waterloo（滑铁卢大学）；Quantum Computing Research Centre（量子计算研究中心）；Hon Hai (Foxconn) Research Institute（鸿海（富士康）研究院）
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 consider a generalization of the standard oracle model in which the oracle acts on the target with a permutation selected according to internal random coins. We describe several problems that are impossible to solve classically but can be solved by a quantum algorithm using a single query; we show that such infinity-vs-one separations between classical and quantum query complexities can be constructed from much weaker separations. We also give conditions to determine when oracle problems---either in the standard model, or in any of the generalizations we consider---cannot be solved with suc…
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.
We study the ability of efficient quantum verifiers to decide properties of exponentially large subsets given either a classical or quantum witness. We develop a general framework that can be used to prove that QCMA machines, with only classical witnesses, cannot verify certain properties of subsets given implicitly via an oracle. We use this framework to prove an oracle separation between QCMA and QMA using an "in-place" permutation oracle, making the first progress on this question since Aaronson and Kuperberg in 2007. We also use the framework to prove a particularly simple standard oracle …
A quantum circuit to construct all maximal cliques using Grover search algorithm is presented. This oracle circuit takes as input an n-qubit state |x> and the adjacency matrix data A of an n-node network, and outputs the state (-1)^f(x) |x> where f(x)=1 if |x> is a maximal clique, and f(x)=0, otherwise. The oracle workspace requires 2n^2 qubits. Each oracle call makes about 10n^2 calls to the Toffoli gates. The overall Grover search algorithm takes as input a uniform superposition n-qubit state and the adjacency matrix data, and outputs a uniform superposition of all maximal clique states.
We introduce the problem of unitarization. Unitarization is the problem of taking $k$ input quantum circuits that produce orthogonal states from the all $0$ state, and create an output circuit implementing a unitary with its first $k$ columns as those states. That is, the output circuit takes the $k$th computational basis state to the state prepared by the $k$th input circuit. We allow the output circuit to use ancilla qubits initialized to $0$. But ancilla qubits must always be returned to $0$ for any input. The input circuits may use ancilla qubits, but we are only guaranteed the they return…
We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations and thus reformulate the quantum walk in a way that circumvents the implementation of that specific oracle and which closely follows the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our num…
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）
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…
A delegation scheme allows a computationally weak client to use a server's resources to help it evaluate a complex circuit without leaking any information about the input (other than its length) to the server. In this paper, we consider delegation schemes for quantum circuits, where we try to minimize the quantum operations needed by the client. We construct a new scheme for delegating a large circuit family, which we call "C+P circuits". "C+P" circuits are the circuits composed of Toffoli gates and diagonal gates. Our scheme is non-interactive, requires very little quantum computation from th…
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…
A conjecture of Jozsa (arXiv:quant-ph/0508124) states that any polynomial-time quantum computation can be simulated by polylogarithmic-depth quantum computation interleaved with polynomial-depth classical computation. Separately, Aaronson conjectured that there exists an oracle $\mathcal{O}$ such that $\textrm{BQP}^{\mathcal{O}} \neq (\textrm{BPP}^\textrm{BQNC})^{\mathcal{O}}$. These conjectures are intriguing allusions to the unresolved potential of combining classical and low-depth quantum computation. In this work we show that the Welded Tree Problem, which is an oracle problem that can be …
We study the computational power of shallow quantum circuits with $O(\log n)$ initialized and $n^{O(1)}$ uninitialized ancillary qubits, where $n$ is the input length and the initial state of the uninitialized ancillary qubits is arbitrary. First, we show that such a circuit can compute any symmetric function on $n$ bits that is classically computable in polynomial time. Then, we regard such a circuit as an oracle and show that a polynomial-time classical algorithm with the oracle can estimate the elements of any unitary matrix corresponding to a constant-depth quantum circuit on $n$ qubits. S…
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, …
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 study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using $\tilde{O}(1)$ quantum queries to a membership oracle, which is an exponential quantum speed-up over the $Ω(n)$ membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz …
Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum …
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.
From Leaves to Clusters: Depth-Efficient SAT-Oracle Synthesis Based on the HRSE Model
Laboratory for Advanced Computing and Intelligence Engineering（先进计算与智能工程实验室）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area（粤港澳大湾区量子科学中心）；Hangzhou Dianzi University（杭州电子科技大学）
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.
