Suppose a Boolean function $f$ is symmetric under a group action $G$ acting on the $n$ bits of the input. For which $G$ does this mean $f$ does not have an exponential quantum speedup? Is there a characterization of how rich $G$ must be before the function $f$ cannot have enough structure for quantum algorithms to exploit?
In this work, we make several steps towards understanding the group actions $G$ which are "quantum intolerant" in this way. We show that sufficiently transitive group actions do not allow a quantum speedup, and that a "well-shuffling" property of group actions -- which happens to be preserved by several natural transformations -- implies a lack of super-polynomial speedups for functions symmetric under the group action. Our techniques are motivated by a recent paper by Chailloux (2018), which deals with the case where $G=S_n$.
Our main application is for graph symmetries: we show that any Boolean function $f$ defined on the adjacency matrix of a graph (and symmetric under relabeling the vertices of the graph) has a power $6$ relationship between its randomized and quantum query complexities, even if $f$ is a partial function. In particular, this means no graph property testing problems can have super-polynomial quantum speedups, settling an open problem of Ambainis, Childs, and Liu (2011).
It has long been known that in the usual black-box model, one cannot get super-polynomial quantum speedups without some promise on the inputs. In this paper, we examine certain types of symmetric promises, and show that they also cannot give rise to super-polynomial quantum speedups. We conclude that exponential quantum speedups only occur given "structured" promises on the input. Specifically, we show that there is a polynomial relationship of degree $12$ between $D(f)$ and $Q(f)$ for any function $f$ defined on permutations (elements of $\{0,1,\dots, M-1\}^n$ in which each alphabet element o…
Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum s…
In this paper, we report the first quantum key-recovery attack on a symmetric block cipher design, using classical queries only, with a more than quadratic time speedup compared to the best classical attack. We study the 2XOR-Cascade construction of Gaži and Tessaro (EUROCRYPT~2012). It is a key length extension technique which provides an n-bit block cipher with 5n/2 bits of security out of an n-bit block cipher with 2n bits of key, with a security proof in the ideal model. We show that the offline-Simon algorithm of Bonnetain et al. (ASIACRYPT~2019) can be extended to, in particular, attack …
Tunneling is often claimed to be the key mechanism underlying possible speedups in quantum optimization via quantum annealing (QA), especially for problems featuring a cost function with tall and thin barriers. We present and analyze several counterexamples from the class of perturbed Hamming-weight optimization problems with qubit permutation symmetry. We first show that, for these problems, the adiabatic dynamics that make tunneling possible should be understood not in terms of the cost function but rather the semi-classical potential arising from the spin-coherent path integral formalism. W…
Bob hides a ball in one of four drawers. Alice is to locate it. Classically she has to open up to three drawers, quantally just one. The fundamental reason for this quantum speedup is not known. The usual representation of the quantum algorithm is limited to the process of solving the problem. We extend it to the process of setting the problem. The number of the drawer with the ball becomes a unitary transformation of the random outcome of the preparation measurement. This extended, time-symmetric, representation brings in relational quantum mechanics. It is with respect to Bob and any externa…
The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum computation: the problem-setting and problem solution, in their quantum version, constitu…
It is shown in the paper that the unitary quantum dynamics in quantum mechanics is the universal quantum driving force to speed up a quantum computation. This assertion supports strongly in theory that the unitary quantum dynamics is the fundamental and universal principle in nature. On the other hand, the symmetric structure of Hilbert space of a composite quantum system is the quantum-computing resource that is not owned by classical computation. A new quantum-computing speedup theory is set up on the basis of the unitary quantum dynamics. Both the unitary quantum dynamics and the symmetric …
We create a variety of new quantum algorithms that use Grover's algorithm and similar techniques to give polynomial speedups over their classical counterparts. We begin by introducing a set of tools that carefully minimize the impact of errors on running time; those tools provide us with speedups to already-published quantum algorithms, such as improving Durr, Heiligman, Hoyer and Mhalla's algorithm for single-source shortest paths [quant-ph/0401091] by a factor of lg N. The algorithms we construct from scratch have a range of speedups, from O(E)->O(sqrt(VE lg V)) speedups in graph theory to a…
Subspace-selective unitary manipulation based on the Hilbert-space symmetric structures in the multiple-quantum operator algebra spaces in the quantum-computing speedup theory
We introduce nebit, a classical bit with a signed probability distribution. We quantify its entropy using Szekely theorem on signed probability measures and show that classical stochastic dynamics supplemented with nebits can achieve or even exceed the speedup of Grover's quantum search algorithm. The proposed classical dynamics never reveals negativity of nebits and thus we do not need any operational interpretation of negative probabilities. We argue that nebits can be useful as a measure of non-classicality as well as a tool to find new quantum algorithms.
A unitary dynamical state-locking (UNIDYSLOCK) process is a unitary process that can transform two orthogonal states to two non-orthogonal states. Its inverse process, i.e., the QUANSDAM process could realize an exponential QC speedup in solving an unstructured search problem in the quantum-computing (QC) speedup theory (X. Miao, arXiv:1105.3573 [quant-ph] (2011)). In this paper the principle of how a UNIDYSLOCK process works is described on the basis of the QC speedup theory. A UNIDYSLOCK process does not exist at all in pure quantum mechanics. That it can change the quantum-state difference …
We propose a variational quantum attack algorithm (VQAA) for classical AES-like symmetric cryptography, as exemplified the simplified-data encryption standard (S-DES). In the VQAA, the known ciphertext is encoded as the ground state of a Hamiltonian that is constructed through a regular graph, and the ground state can be found using a variational approach. We designed the ansatz and cost function for the S-DES's variational quantum attack. It is surprising that sometimes the VQAA is even faster than Grove's algorithm as demonstrated by our simulation results. The relationships of the entanglem…
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 …
Precise and efficient control of quantum systems is essential to perform quantum information processing tasks. In terms of adiabatic speedup via leakage elimination operator approach, for a closed system, the ideal pulse control conditions have been theoretically derived by P-Q partitioning technique. However, it is a challenge to design the corresponding control pulses for an open system, which requires to tackle noisy environments. In this paper, we apply the stochastic search procedures to an open qutrit system and successfully obtain the optimal control pulses for significant adiabatic spe…
Understanding heat transfer between a quantum system and its environment is of undisputed importance if reliable quantum devices are to be constructed. Here, we investigate the heat transfer between system and bath in non-Markovian open systems in the process of adiabatic speedup. Using the quantum state diffusion equation method, the heat current, energy current, and power are calculated during free evolution and under external control of the system. While the heat current increases with increasing system-bath coupling strength and bath temperature, it can be restricted by the non-Markovian n…
Motivated by the quantum speedup for dynamic programming on the Boolean hypercube by Ambainis et al. (2019), we investigate which graphs admit a similar quantum advantage. In this paper, we examine a generalization of the Boolean hypercube graph, the $n$-dimensional lattice graph $Q(D,n)$ with vertices in $\{0,1,\ldots,D\}^n$. We study the complexity of the following problem: given a subgraph $G$ of $Q(D,n)$ via query access to the edges, determine whether there is a path from $0^n$ to $D^n$. While the classical query complexity is $\widetildeΘ((D+1)^n)$, we show a quantum algorithm with compl…
It is often argued that entanglement is at the root of the speedup for quantum compared to classical computation, and that one needs a sufficient amount of entanglement for this speedup to be manifest. In measurement-based quantum computing (MBQC), the need for a highly entangled initial state is particularly obvious. Defying this intuition, we show that quantum states can be too entangled to be useful for the purpose of computation. We prove that this phenomenon occurs for a dramatic majority of all states: the fraction of useful n-qubit pure states is less than exp(-n^2). Computational unive…
Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk's hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk's random sampling yields an arbitrary speedup in query complexity over the classical random …
One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, …
Can quantum mechanics help us in building intelligent robots and agents? One of the defining characteristics of intelligent behavior is the capacity to learn from experience. However, a major bottleneck for agents to learn in any real-life situation is the size and complexity of the corresponding task environment. Owing to, e.g., a large space of possible strategies, learning is typically slow. Even for a moderate task environment, it may simply take too long to rationally respond to a given situation. If the environment is impatient, allowing only a certain time for a response, an agent may t…
We study scattering quantum walks on highly symmetric graphs and use the walks to solve search problems on these graphs. The particle making the walk resides on the edges of the graph, and at each time step scatters at the vertices. All of the vertices have the same scattering properties except for a subset of special vertices. The object of the search is to find a special vertex. A quantum circuit implementation of these walks is presented in which the set of special vertices is specified by a quantum oracle. We consider the complete graph, a complete bipartite graph, and an $M$-partite graph…
We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this problem. Furthermore, quantum Schur transforms yield efficient solutions for certain irreducible representations of the unitary group. Beyond this, we obtain poly(n)-time quantum algorithms for approximating matrix elements from all the irreducible representations of the alternating group A_n, and all the irreducible representations of polynomial highest weight of …
Shenzhen Institute for Quantum Science and Engineering（深圳量子科学与工程研究院）；University of Science and Technology, Shenzhen（深圳理工大学）；University of Technology Sydney（悉尼科技大学）；Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Central Research Institute, Huawei Technologies（华为技术有限公司中央研究院）
In classical machine learning, a set of weak classifiers can be adaptively combined to form a strong classifier for improving the overall performance, a technique called adaptive boosting (or AdaBoost). However, constructing the strong classifier for a large data set is typically resource consuming. Here we propose a quantum extension of AdaBoost, demonstrating a quantum algorithm that can output the optimal strong classifier with a quadratic speedup in the number of queries of the weak classifiers. Our results also include a generalization of the standard AdaBoost to the cases where the outpu…
Solving eigenproblem of the Laplacian matrix of a fully connected weighted graph has wide applications in data science, machine learning, and image processing, etc. However, this is very challenging because it involves expensive matrix operations. Here, we propose an efficient quantum algorithm to solve it based on a assumption that the element of each vertex and its norms can be effectively accessed via a quantum random access memory data structure. Specifically, we adopt the optimal Hamiltonian simulation technique based on the block-encoding framework to implement the quantum simulation of …
Lin and Lin have recently shown how starting with a classical query algorithm (decision tree) for a function, we may find upper bounds on its quantum query complexity. More precisely, they have shown that given a decision tree for a function $f:\{0,1\}^n\to[m]$ whose input can be accessed via queries to its bits, and a guessing algorithm that predicts answers to the queries, there is a quantum query algorithm for $f$ which makes at most $O(\sqrt{GT})$ quantum queries where $T$ is the depth of the decision tree and $G$ is the maximum number of mistakes of the guessing algorithm. In this paper w…
If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be un…
Quantum computers will be able solve important problems with significant polynomial and exponential speedups over their classical counterparts, for instance in option pricing in finance, and in real-space molecular chemistry simulations. However, key applications can only achieve their potential speedup if their inputs are prepared efficiently. We effectively solve the important problem of efficiently preparing quantum states following arbitrary continuous (as well as more general) functions with complexity logarithmic in the desired resolution, and with rigorous error bounds. This is enabled …
We formulate a framework for discrete-time quantum walks, motivated by classical random walks with memory. We present a specific representation of the classical walk with memory 2 on which this is based. The framework has no need for coin spaces, it imposes no constraints on the evolution operator other than unitarity, and is unifying of other approaches. As an example we construct a symmetric discrete-time quantum walk on the semi-infinite binary tree. The generating function of the amplitude at the root is computed in closed-form, as a function of time and the initial level n in the tree, an…
We consider the special case of Random Tensor Networks (RTN) endowed with gauge symmetry constraints on each tensor. We compute the Rènyi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background independe…
