A Factorisation Algorithm in Adiabatic Quantum Computation

The problem of factorising positive integer $N$ into two integer factors $x$ and $y$ is first reformulated as an optimisation problem over the positive integer domain of either of the Diophantine polynomials $Q_N(x,y)=N^2(N-xy)^2 + x(x-y)^2$ or $R_N(x,y) = N^2(N-xy)^2 + (x-y)^2 + x$, of each of which the optimal solution is unique with $x\le \sqrt{N} \le y$, and $x=1$ if and only if $N$ is prime. An algorithm in the context of Adiabatic Quantum Computation is then proposed for the general factorisation problem.

Factoring an integer into its prime constituents has attracted much interest since the advance of the RSA public-private key encryption [1]. It is suspected that factorisation is NP-intermediate, that is, in the NP class but may be not quite NP-complete. While there does not yet exist any polynomial-time algorithm for the problem on a classical/Turing computer, the discovery of Shor's quantum algorithm with quantum circuits [2] has been one of the main motivations for research into quantum computation and building of quantum computers.
In this paper we also consider the factorisation problem in the realm of quantum computation but with adiabatic processes, in complementary addition to the computation with quantum circuits. Also recently, the authors of [3] have considered the problem with quantum annealing. We first reformulate in the next section the factorisation into two integer factors as an optimisation of some corresponding Diophantine polynomial over the integer domain. The optimisation could also be repeatedly applied to any integer having more than two prime factors. Based on this reformulation, we then present an algorithm in the context of AQC (Adiabatic Quantum Computation) for the general factorisation problem. Following that are some numerical illustrations of the algorithm and discussion on the lower bound of the computing time with the help of an energy-time uncertainty relation. The paper is then concluded with some remarks.

FACTORISATION AS AN OPTIMISATION PROBLEM
We first consider the problem of factorising a natural integer N into two integer factors x and y. We propose that this problem can be reformulated as an optimisation problem over the integer domain of the following Diophantine polynomial Without the second term in Q N above, the optimal solutions contain the trivial unity factor.
We could eliminate this triviality with this second term replaced by (x − y) 2 , but there still remains a symmetry between x and y. We now show that the polynomial Q N (x, y) in (1) is minimised if and only if xy = N where x ≤ y and x is nearest to, but not exceeding, the integer part of √ N . The optimal solution thus has no trivial factor x = 1 unless N is prime.
For xy = N the first term on the rhs of (1) vanishes and the remaining second term is obviously smaller for 1 ≤ x ≤ √ N ≤ y. In this case, the second term is a one-variable function x(x − N/x) 2 , and it is not difficult to see that this term is a decreasing function for The second line results from the fact that x(x − N/x) 2 obtains its maximum value at the boundary where x = 1. The closer x is to the integer part of √ N the smaller the value of Now if xy = N then obviously (N − xy) 2 ≥ 1 and the first term of (1) is consequently not less than N 2 . Thus Combining (2) and (3), we have for N > 1 The second term x(x − y) 2 of Q N (x, y) in (1) is thus designed to introduce an asymmetry between x and y, enforcing x ≤ y at the optimum value, and to eliminate the trivial unity factor of N unless N is prime. The optimisation problem (1) thus has a unique solution (x, y) for xy = N with x ≤ √ N ≤ y. In general, x is the integer factor nearest to, but not exceeding, √ N . Consequently, x = 1 if and only if N is prime.
The optimisation of Q N (x, y) can thus also determine the primality of N . For N having more than two prime factors we can apply the optimisation repeatedly for the successive factors to obtain the constituent prime factors.
It can be shown that the objective function in (1) is non-convex as depicted in FIG. 1 for a slice of the polynomial for N = 6 along the plane y + x = 5. (In general, determining convexity for multivariate quartic polynomials is NP-hard.) It is also known that the problem of minimising a general non-convex degree four polynomial over a two-dimensional convex polygon like that in (1) either in a temporally linear manner (that is, f (t/T ) = (1 − t/T ) and g(t/T ) = t/T ); or otherwise with f (0) = 1 = g(1) and f (1) = 0 = g(0). We also assume that both f and g |ψ(0) = |g I .
As the rate of the evolution of the Hamiltonian approaches zero, the end state |ψ(T ) asymptotically converges to the target state |g as asserted by the quantum adiabatic theorem [7].
The initial Hamiltonian H I and its ground state |g I Corresponding to the variable x, we introduce the operators a † x and a x , which respectively create and annihilate the number states |n x , for n x = 0, 1, . . .: a † x |n x = √ n x + 1|n x + 1 , and the number operatorsn x are constructed in the usual way: Similarly for the variable y with a † y , a y andn y . We propose to start the AQC with the following initial Hamiltonian H I : for some c-numbers θ x and θ y . The states in the total Hilbert space can be now decomposed in terms of |n x ⊗|m y , the direct products of the two bases of number states. This Hamiltonian admits the readily constructible direct product of coherent states as the ground state |g I , with |θ the canonical quantum coherent state where: a |θ = θ |θ .

The target Hamiltonian H P
Building on (1), we introduce the target Hamiltonian H P ,  This would be helpful in shortening the running time with an appropriately chosen g(t/T ) for the AQC algorithm, in accordance with (13) in the next Section.
FIG . 3 shows the probabilities | ψ(T )|n x m y | 2 versus T for the two largest probabilities. In all instances of T there, the number state |2 x 3 y has the maximum probability among all the number states. This probability increases with T , the inverse of the evolution rate, in agreement with the quantum adiabatic theorem [7]. The next highest probability corresponds to the other solution |3 x 2 y .
We note that the increased probability for |3 x 2 y at T = 100 compared to that at T = 30 is an artifact of our truncation of the Hilbert space. With a truncation at |n max such that a † |n max = 0, the commutation relation (7) is violated n max |aa † |n max = 0, n max |a † a + 1|n max = n max + 1.
The effect of such a truncation artifact becomes more prominent for larger T , unless we increase n max accordingly.

COMPUTATIONAL COMPLEXITY
We have presented elsewhere [8] a necessary condition for a lower time limit required in a general AQC for an initial state to evolve into an orthogonal state under the dynamics of H. Namely, it is necessary that the evolution time cannot be less than T ⊥ , where ∆ I E P is the energy spread of the initial state |g I in terms of the target Hamiltonian H P , It is important to note that only the initial eigenstate |g I (which is of course timeindependent), and neither the instantaneous eigenstates nor the full time-dependent wave function at any other times, is required for the time condition (13). This hallmark of our results in [8] enables their wider applicability and usefulness.
We note that T ⊥ is particularly a function of the parameters |θ|, The parameters θ's give our proposed AQC algorithm some advantage that is denied or not evident elsewhere, and this could be exploited in order to reduce the lower time limit We expect that, as there is no reason to the contrary, the lower time limit (13) could be saturated by appropriate choices of the extrapolation functions f (t) and g(t) for H(t) in (5). It is not that surprising that we could reduce the computation time with more energy resources. Other examples of this delicate balance between the energy required and the lower time limit for a well known quantum algorithm are given elsewhere [8,9]. We thus expect that the factorisation problem could be solved efficiently with AQC. This agrees with the fact that the factorisation problem is in the BQP class as also demonstrated by Shor's algorithm for quantum circuits [2].

CONCLUDING REMARKS
We have mapped the factorisation problem into the optimisation over the integer domain of some corresponding Diophantine polynomials. The optimal solutions are unique, and contain the unity factor if and only if the integers to be factored are prime. Based on this, we then propose an algorithm in the context of Adiabatic Quantum Computation for the general factorisation. We expect the running time is polynomial or even less, at the expense of more energy input in order to carry out the physical computation. Some numerical simulations of the algorithm for a simple case are also presented for illustration purpose.
We are grateful for discussions with Peter Hannaford, Adolfo del Campo and Richard Warren, who also brought the work [3] to our attention after the completion of this paper.