Abstract The noisy binary linear problem (NBLP) is known as a computationally hard problem, and therefore, it offers primitives for post-quantum cryptography. An efficient quantum NBLP algorithm that exhibits a polynomial quantum sample and time complexities has recently been proposed. However, the algorithm requires a large number of samples to be loaded in a highly entangled state and it is unclear whether such a precondition on the quantum speedup can be obtained efficiently. Here, we present a complete analysis of the quantum solvability of the NBLP by considering the entire algorithm process, namely from the preparation of the quantum sample to the main computation. By assuming that the algorithm runs on ‘fault-tolerant’ quantum circuitry, we introduce a reasonable measure of the computational time cost. The measure is defined in terms of the overall number of T gate layers, referred to as T-depth complexity. We show that the cost of solving the NBLP can be polynomial in the problem size, at the expense of an exponentially increasing logical qubits.
Abstract The noisy binary linear problem (NBLP) is known as a computationally hard problem, and therefore, it offers primitives for post-quantum cryptography. An efficient quantum NBLP algorithm that exhibits a polynomial quantum sample and time complexities has recently been proposed. However, the algorithm requires a large number of samples to be loaded in a highly entangled state and it is unclear whether such a precondition on the quantum speedup can be obtained efficiently. Here, we present a complete analysis of the quantum solvability of the NBLP by considering the entire algorithm process, namely from the preparation of the quantum sample to the main computation. By assuming that the algorithm runs on ‘fault-tolerant’ quantum circuitry, we introduce a reasonable measure of the computational time cost. The measure is defined in terms of the overall number of T gate layers, referred to as T-depth complexity. We show that the cost of solving the NBLP can be polynomial in the problem size, at the expense of an exponentially increasing logical qubits.
The noisy binary linear problem (NBLP) is a known computationally intractable problem. Thus, NBLP offers primitives for post-quantum cryptography. An efficient quantum NBLP algorithm that exhibits a polynomial quantum sample and time complexities has recently been proposed. However, a large number of samples should be loaded in a highly entangled state, and it is unclear whether such a precondition does not affect the quantum speedup obtained. Here, we analyse the quantum solvability of NBLP by considering the entire algorithm process, namely from the preparation of the quantum sample to the main computation. Assuming that the algorithm runs on fault-tolerant quantum circuitry, the cost is defined in terms of the overall number of layers of T gates, often referred to as T-depth complexity. We show that the cost of solving the NBLP can be polynomial in the problem size, at the expense of an exponentially increasing number of logical qubits.
Abstract Noisy linear problems have been studied in various science and engineering disciplines. A class of ‘hard’ noisy linear problems can be formulated as follows: Given a matrix A ^ and a vector b constructed using a finite set of samples, a hidden vector or structure involved in b is obtained by solving a noise-corrupted linear equation A ^ x ≈ b + η , where η is a noise vector that cannot be identified. For solving such a noisy linear problem, we consider a quantum algorithm based on a divide-and-conquer strategy, wherein a large core process is divided into smaller subprocesses. The algorithm appropriately reduces both the computational complexities and size of a quantum sample. More specifically, if a quantum computer can access a particular reduced form of the quantum samples, polynomial quantum-sample and time complexities are achieved in the main computation. The size of a quantum sample and its executing system can be reduced, e.g., from exponential to sub-exponential with respect to the problem length, which is better than other results we are aware. We analyse the noise model conditions for such a quantum advantage, and show when the divide-and-conquer strategy can be beneficial for quantum noisy linear problems.
Noisy linear problems have been studied in various science and engineering disciplines. A class of hard noisy linear problems can be formulated as follows: Given a matrix A^ and a vector b constructed using a finite set of samples, a hidden vector or structure involved in b is obtained by solving a noise-corrupted linear equation A^x≈b+η, where η is a noise vector that cannot be identified. For solving such a noisy linear problem, we consider a quantum algorithm based on a divide-and-conquer strategy, wherein a large core process is divided into smaller subprocesses. The algorithm appropriately reduces both the computational complexities and size of a quantum sample. More specifically, if a quantum computer can access a particular reduced form of the quantum samples, polynomial quantum-sample and time complexities are achieved in the main computation. The size of a quantum sample and its executing system can be reduced, e.g., from exponential to sub-exponential with respect to the problem length, which is better than other results we are aware. We analyse the noise model conditions for such a quantum advantage, and show when the divide-and-conquer strategy can be beneficial for quantum noisy linear problems.
A defining feature in the field of quantum computing is the potential of a quantum device to outperform its classical counterpart for a specific computational task.By now, several proposals exist showing that certain sampling problems can be done efficiently quantumly, but are not possible efficiently classically, assuming strongly held conjectures in complexity theory, a feature dubbed quantum speedup.However, the effect of noise on these proposals is not well understood in general, and in certain cases it is known that simple noise can destroy the quantum speedup.Here we develop a fault-tolerant version of one family of these sampling problems, based on nonadaptive measurements on regular graph states composed of m qubits.We show that these sampling problems can be implemented fault tolerantly using quantum circuits of constant depth.We present two constructions, each taking poly(m) physical qubits, some of which are prepared in noisy magic states.The first of our constructions is a constant depth quantum circuit composed of single-and two-qubit nearest-neighbor Clifford gates in four dimensions.This circuit has one layer of interaction with a classical computer before final measurements.Our second construction is a constant depth quantum circuit with single-and two-qubit nearest-neighbor Clifford gates in three dimensions, but with two layers of interaction with a classical computer before the final measurements.For each of these constructions, we show that there is no classical algorithm which can sample according to its output distribution in poly(m) time, assuming two standard complexity theoretic conjectures hold.The noise model we assume is the so-called local stochastic quantum noise.Along the way, we introduce various concepts such as constant depth magic state distillation and constant depth output routing, which arise naturally in measurement based quantum computation, but have no constant-depth analog in the circuit model.
Solving combinatorial optimization problems on near-term quantum computers mainly relies on their quadratic unconstrained binary formulation. Lately, binary polynomial formulations start to take the center stage, having a better scaling with respect to the number of used qubits. However, while quadratic forms are straightforward to implement on quantum circuits, polynomial forms of degree d incur in circuit depth overheads, due to the lack of d-qubit gates, for d≥3, on current hardware. In this paper, we study the problem of designing the shallowest quantum circuit for a given polynomial form. We formulate it as a classical binary linear problem and we prove the \textsf{NP}-completeness of a variant of it. On the algorithmic side, we offer a scalable and performant heuristic method to tackle it, Tempura, as well as faster binary linear problem formulations, which offer computational feasibility while sacrificing optimality. Our performance evaluation shows that the algorithms yield far shorter circuits than the state-of-the-art approaches, and Tempura is mostly comparable with them in terms of runtime.
Very recently, one of the most biggest agenda issues is to provide the proof of quantum computational speedup, particularly with a prospect for near-term uses. However, many quantum algorithms are beyond the reach of noisy intermediate-scale quantum (NISQ) realization. This is mainly because of the requirement of excessively large superposition and massive quantum circuit. Hence, we propose a (say) ``NISQ-compatible'' algorithm for one of the crucial problems in computation and modern cryptography, the learning-with-errors (LWE) problem. We base an approach on the divide-and-conquer, wherein a large core process is subdivided into smaller subprocesses. For a specific error distribution and problem condition, it is shown that the proposed quantum LWE algorithm allows the use of exponentially less-superposed quantum samples and operation overheads are reduced, while achieving polynomial sample complexity.