Simon’s Algorithm

Overview

This module emulates Simon’s Algorithm.

Simon’s problem is summarized as follows. A function \(f :\{0,1\}^n\rightarrow\{0,1\}^n\) is promised to be either one-to-one, or two-to-one with some nonzero \(n\)-bit mask \(s\). The latter condition means that for any two different \(n\)-bit numbers \(x\) and \(y\), \(f(x)=f(y)\) if and only if \(x\oplus y = s\). The problem then is to determine whether \(f\) is one-to-one or two-to-one, and, if the latter, what the mask \(s\) is, in as few queries to \(f\) as possible.

The problem statement and algorithm can be explored further, at a high level, in reference [1]. The implementation of the algorithm in this module, however, follows [2].

Algorithm and Details

This algorithm involves a quantum component and a classical component. The quantum part follows similarly to other blackbox oracle algorithms. First, assume a blackbox oracle \(U_f\) is available with the property $$U_f\vert x\rangle\vert y\rangle = \vert x\rangle\vert y\oplus f(x)\rangle$$

where the top \(n\) qubits \(\vert x \rangle\) are the input, and the bottom \(n\) qubits \(\vert y \rangle\) are called ancilla qubits.

The input qubits are prepared with the ancilla qubits into the state $$(H^{\otimes n} \otimes I^{\otimes n})\vert 0\rangle^{\otimes n}\vert 0\rangle^{\otimes n} = \vert +\rangle^{\otimes n}\vert 0\rangle^{\otimes n}$$ and sent through a blackbox gate \(U_f\). Then, the Hadamard-Walsh transform \(H^{\otimes n}\) is applied to the \(n\) input qubits, resulting in the state given by

$$(H^{\otimes n} \otimes I^{\otimes n})U_f\vert +\rangle^{\otimes n}\vert 0\rangle^{\otimes n}$$

It turns out the resulting \(n\) input qubits are in a uniform random state over the space killed by (modulo \(2\), bitwise) dot product with \(s\). This covers the one-to-one case as well, if one considers it to be the degenerate \(s=0\) case.

Suppose we then measured the \(n\) input qubits, calling the bitstring output \(y\). The above property then requires \(s\cdot y = 0\). The space of \(y\) that satisfies this is \(n-1\) dimensional. By running this quantum subroutine several times, \(n-1\) nonzero linearly independent bitstrings \(y_i\), \(i = 0, \ldots, n-2\), can be found, each orthogonal to \(s\).

This gives a system of \(n-1\) equations, with \(n\) unknowns for finding \(s\). One final nonzero bitstring \(y^{\prime}\) can be classically found that is linearly independent to the other \(y_i\), but with the property that \(s\cdot y^{\prime} = 1\). The combination of \(y^{\prime}\) and the \(y_i\) give a system of \(n\) independent equations that can then be solved for \(s\).

By using a clever implementation of Gaussian Elimination and Back Substitution for mod-2 equations, as outlined in Reference [2], \(s\) can be found relatively quickly. By then sending separate input states \(\vert 0\rangle\) and \(\vert s\rangle\) through the blackbox \(U_f\), we can find whether or not \(f(0) = f(s)\) (in fact, any pair \(\vert x\rangle\) and \(\vert x\oplus s\rangle\) will do as well). If so, we conclude \(f\) is two-to-one with mask \(s\); otherwise, \(f\) is one-to-one.

Overall, this algorithm can be solved in \(O(n^3)\), i.e., polynomial, time, whereas the best classical algorithm requires exponential time.

Source Code Docs

Here you can find documentation for the different submodules in phaseestimation.