Quantum optimization algorithms

One of the most common types of data fitting is solving the least squares problem, minimizing the sum of the squares of differences between the data points and the fitted function.

The algorithm is given ${\displaystyle N}$ input data points ${\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),...,(x_{N},y_{N})}$ and ${\displaystyle M}$ continuous functions ${\displaystyle f_{1},f_{2},...,f_{M}}$. The algorithm finds and gives as output a continuous function ${\displaystyle f_{\vec {\lambda }}}$ that is a linear combination of ${\displaystyle f_{j}}$:

${\displaystyle f_{\vec {\lambda }}(x)=\sum _{j=1}^{M}f_{j}(x)\lambda _{j}}$

In other words, the algorithm finds the complex coefficients ${\displaystyle \lambda _{j}}$, and thus the vector ${\displaystyle {\vec {\lambda }}=(\lambda _{1},\lambda _{2},...,\lambda _{M})}$.

The algorithm is aimed at minimizing the error, which is given by:

${\displaystyle E=\sum _{i=1}^{N}\left\vert f_{\vec {\lambda }}(x_{i})-y_{i}\right\vert ^{2}=\sum _{i=1}^{N}\left\vert \sum _{j=1}^{M}f_{j}(x_{i})\lambda _{j}-y_{i}\right\vert ^{2}=\left\vert F{\vec {\lambda }}-{\vec {y}}\right\vert ^{2}}$

where ${\displaystyle F}$ is defined to be the following matrix:

${\displaystyle {F}={\begin{pmatrix}f_{1}(x_{1})&\cdots &f_{M}(x_{1})\\f_{1}(x_{2})&\cdots &f_{M}(x_{2})\\\vdots &\ddots &\vdots \\f_{1}(x_{N})&\cdots &f_{M}(x_{N})\\\end{pmatrix}}}$

The quantum least-squares fitting algorithm[2] makes use of a version of Harrow, Hassidim, and Lloyd's quantum algorithm for linear systems of equations (HHL), and outputs the coefficients ${\displaystyle \lambda _{j}}$ and the fit quality estimation ${\displaystyle E}$. It consists of three subroutines: an algorithm for performing a pseudo-inverse operation, one routine for the fit quality estimation, and an algorithm for learning the fit parameters.

Because the quantum algorithm is mainly based on the HHL algorithm, it suggests an exponential improvement[3] in the case where ${\displaystyle F}$ is sparse and the condition number (namely, the ratio between the largest and the smallest eigenvalues) of both ${\displaystyle FF^{\dagger }}$ and ${\displaystyle F^{\dagger }F}$ is small.

The algorithm inputs are ${\displaystyle A_{1}...A_{m},C,b_{1}...b_{m}}$ and parameters regarding the solution's trace, precision and optimal value (the objective function's value at the optimal point).

The quantum algorithm[5] consists of several iterations. In each iteration, it solves a feasibility problem, namely, finds any solution satisfying the following conditions (giving a threshold ${\displaystyle t}$):

${\displaystyle {\begin{array}{lr}\langle C,X\rangle _{\mathbb {S} ^{n}}\leq t\\\langle A_{k},X\rangle _{\mathbb {S} ^{n}}\leq b_{k},\quad k=1,\ldots ,m\\X\succeq 0\end{array}}}$

In each iteration, a different threshold ${\displaystyle t}$ is chosen, and the algorithm outputs either a solution ${\displaystyle X}$ such that ${\displaystyle \langle C,X\rangle _{\mathbb {S} ^{n}}\leq t}$ (and the other constraints are satisfied, too) or an indication that no such solution exists. The algorithm performs a binary search to find the minimal threshold ${\displaystyle t}$ for which a solution ${\displaystyle X}$ still exists: this gives the minimal solution to the SDP problem.

The quantum algorithm provides a quadratic improvement over the best classical algorithm in the general case, and an exponential improvement when the input matrices are of low rank.

For combinatorial optimization, the quantum approximate optimization algorithm (QAOA)[6] briefly had a better approximation ratio than any known polynomial time classical algorithm (for a certain problem),[7] until a more effective classical algorithm was proposed.[8] The relative speed-up of the quantum algorithm is an open research question.

The heart of QAOA relies on the use of unitary operators dependent on ${\displaystyle 2p}$ angles, where ${\displaystyle p>1}$ is an input integer. These operators are iteratively applied on a state that is an equal-weighted quantum superposition of all the possible states in the computational basis. In each iteration, the state is measured in the computational basis and ${\displaystyle C(z)}$ is estimated. The angles are then updated classically to increase ${\displaystyle C(z)}$. After this procedure is repeated a sufficient number of times, the value of ${\displaystyle C(z)}$ is almost optimal, and the state being measured is close to being optimal as well. In principle the optimal value of ${\displaystyle C(z)}$ can be reached up to arbitrary precision, this is guaranteed by the adiabatic theorem[9] or alternatively by the universality of the QAOA unitaries.[10] However, it is an open question whether this can be done in a feasible way. For example, it was shown that QAOA exhibits a strong dependence on the ratio of a problem's constraint to variables (problem density) placing a limiting restriction on the algorithm's capacity to minimize a corresponding objective function.[11]

It was soon recognized that a generalization of the QAOA process is essentially an alternating application of a continuous-time quantum walk on an underlying graph followed by a quality-dependent phase shift applied to each solution state. This generalized QAOA was termed as QWOA (Quantum Walk-based Optimisation Algorithm).[12]

In the paper How many qubits are needed for quantum computational supremacy submitted to arXiv,[13] the authors conclude that a QAOA circuit with 420 qubits and 500 constraints would require at least one century to be simulated using a classical simulation algorithm running on state-of-the-art supercomputers so that would be sufficient for quantum computational supremacy.

A rigorous comparison of QAOA with classical algorithms can give estimates on depth ${\displaystyle p}$ and number of qubits required for quantum advantage. A study of QAOA and MaxCut algorithm shows that ${\displaystyle p>11}$ is required for scalable advantage.[14]