Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms. Shor's quantum algorithm for fast number factoring is a key example and the prime motivator in the international effort to realise a quantum computer. However, due to the substantial resource requirement, to date, there have been only four small-scale demonstrations. Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which the n qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol. Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N=21. The algorithmic output is distinguishable from noise, in contrast to previous demonstrations. These results point to larger-scale implementations of Shor's algorithm by harnessing scalable resource reductions applicable to all physical architectures.
Integer factorization has been one of the cornerstone applications of the field of quantum computing since the discovery of an efficient algorithm for factoring by Peter Shor. Unfortunately, factoring via Shor's algorithm is well beyond the capabilities of today's noisy intermediate-scale quantum (NISQ) devices. In this work, we revisit the problem of factoring, developing an alternative to Shor's algorithm, which employs established techniques to map the factoring problem to the ground state of an Ising Hamiltonian. The proposed variational quantum factoring (VQF) algorithm starts by simplify…
A scheme to encode arbitrarily long integer pairs on degenerate optical parametric oscillations multiplexed in time is proposed. The classical entanglement between the polarization directions and the phases of the oscillating pulses, regarded as two computational registers, furnishes the integer correlations within each pair. We show the major algorithmic steps, modular exponentiation and discrete Fourier transform, of Shor's quantum factoring algorithm can be executed in the registers as pulse interferences under the assistance of external logics. The factoring algorithm is thus rendered equi…
We report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous demonstration by Martín-López et al. in Nature Photonics 6, 773 (2012). We go beyond this work by using a configuration of approximate Toffoli gates with residual phase shifts, which preserves the functional correctness and allows us to achieve a complete factoring of N=21. We implemented the algorithm on IBM quantum processors using only 5 qubits and successfull…
We study the results of a compiled version of Shor's factoring algorithm on the ibmqx5 superconducting chip, for the particular case of $N=15$, $21$ and $35$. The semi-classical quantum Fourier transform is used to implement the algorithm with only a small number of physical qubits and the circuits are designed to reduce the number of gates to the minimum. We use the square of the statistical overlap to give a quantitative measure of the similarity between the experimentally obtained distribution of phases and the predicted theoretical distribution one for different values of the period. This …
We present reversible classical circuits for performing various arithmetic operations aided by dirty ancillae (i.e. extra qubits in an unknown state that must be restored before the circuit ends). We improve the number of clean qubits needed to factor an n-bit number with Shor's algorithm from 1.5n+O(1) to n+2, assisted by n-1 dirty qubits, without increasing the asymptotic size or depth of the circuit.
Experimental factoring integers using fixed-point-QAOA with a trapped-ion quantum processor
P.N. Lebedev Physical Institute of the Russian Academy of Sciences（俄罗斯科学院列别捷夫物理研究所）；Russian Quantum Center（俄罗斯量子中心）；National University of Science and Technology “MISIS”（国立科技大学 MISIS）
The assumed computationally difficulty of factoring large integers forms the basis of security for RSA public-key cryptography, which specifically relies on products of two large primes or semi-primes. The best-known factoring algorithms for classical computers run in sub-exponential time. Since integer factorization is in NP, one can reduce this problem to any NP-hard problem, such as Boolean Satisfiability (SAT). While reducing factoring to SAT has proved to be useful for studying SAT solvers, attempting to factor large integers via such a reduction has not been found to be successful. Shor'…
Shor's factoring algorithm provides a super-polynomial speed-up over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor's algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the prot…
Shor's algorithm has seriously challenged information security based on public key cryptosystems. However, to break the widely used RSA-2048 scheme, one needs millions of physical qubits, which is far beyond current technical capabilities. Here, we report a universal quantum algorithm for integer factorization by combining the classical lattice reduction with a quantum approximate optimization algorithm (QAOA). The number of qubits required is O(logN/loglog N), which is sublinear in the bit length of the integer $N$, making it the most qubit-saving factorization algorithm to date. We demonstra…
There exists a Hamiltonian formulation of the factorisation problem which also needs the definition of a factorisation ensemble (a set to which factorable numbers, $N'=x'y'$, having the same trivial factorisation algorithmic complexity, belong). For the primes therein, a function $E$, that may take only discrete values, should be the analogous of the energy from a confined system of charges in a magnetic trap. This is the quantum factoring simulator hypothesis connecting quantum mechanics with number theory. In this work, we report numerical evidence of the existence of this kind of discrete s…
We analyze the performance of a quantum computer architecture combining a small processor and a storage unit. By focusing on integer factorization, we show a reduction by several orders of magnitude of the number of processing qubits compared with a standard architecture using a planar grid of qubits with nearest-neighbor connectivity. This is achieved by taking advantage of a temporally and spatially multiplexed memory to store the qubit states between processing steps. Concretely, for a characteristic physical gate error rate of $10^{-3}$, a processor cycle time of 1 microsecond, factoring a…
