Prime factorization using quantum variational imaginary time evolution

Abstract

The road to computing on quantum devices has been accelerated by the promises that come from using Shor’s algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and lack of robust error correction schemes. Here we explore a promising, alternative method for prime factorization that uses well-established techniques from variational imaginary time evolution. We create a Hamiltonian whose ground state encodes the solution to the problem and use variational techniques to evolve a state iteratively towards these prime factors. We show that the number of circuits evaluated in each iteration scales as $$O(n^{5}d)$$ O ( n 5 d ) , where n is the bit-length of the number to be factorized and d is the depth of the circuit. We use a single layer of entangling gates to factorize 36 numbers represented using 7, 8, and 9-qubit Hamiltonians. We also verify the method’s performance by implementing it on the IBMQ Lima hardware to factorize 55, 65, 77 and 91 which are greater than the largest number (21) to have been factorized on IBMQ hardware.

Variational Quantum Circuits to Prepare Low Energy Symmetry States

“…The variational minimization over θ is done over many realizations of the parameter set α, which is akin to an averaging procedure over the underlying sampling distribution p( α). Mathematically, the parameter set θ is learned as follows, (5) where p( α) represents the averaging over the distribution p( α). The parameters θ * are trained so as to achieve a very low margin of error of allowing Ũ to faithfully mimic the exact unitary U and confine any subsequent operation to the symmetry subspace Ω irrespective of the state prepared by the ansatz A.…”

“…Phase estimation algorithm [1,2], despite solving this problem, requires circuits that run deep, something that cannot be afforded within the NISQ [3] era. Alternative algorithms based on hybrid models that make use of variational methods, for instance variational quantum eigensolvers (VQE) [4] and quantum imaginary time evolution [5], have been found to be more resilient to noisy quantum devices [6]. The strength of a variational method depends a lot on the variational ansatz that has been used for the simulation of the given state.…”

Psitrum: An Open Source Simulator for Universal Quantum Computers

“…This section is about testing Psitrum to validate its performance by implementing four quantum algorithms, quantum fulladder 58 , Deutsch-Joza 59 , Grover Search 60 and prime factorization 61 algorithms. These circuits are good benchmark problems for a universal quantum computer simulator [62][63][64][65] .…”

Psitrum: An Open Source Simulator for Universal Quantum Computers

“…This section is about testing Psitrum to validate its performance by implementing four quantum algorithms, quantum full-adder [58], Deutsch-Joza [59], Grover Search [60] and prime factorization [61] algorithms. These circuits are good benchmark problems for a universal quantum computer simulator [62,63,64,65].…”

“…The cost function used is given by (N − pq) 2 , where N is the number to be factorized, while p and q are factors to be identified and expressed in binary form as qubits over which the optimization is performed. Refer [61] for a complete discussion on how the cost function is constructed and its complexity. The authors there made use of imaginary time evolution to solve for the factors.…”