Optimal control-based efficient synthesis of building blocks of quantum algorithms: A perspective from network complexity towards time complexity

Abstract

In this paper, we demonstrate that optimal control algorithms can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristic ratio of the time scales for local controls vs non-local (i.e. coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity.In the limit of fast local controls (as e.g. in NMR), time-optimised realisations are shown for the quantum Fourier transform (QFT) and the multiply controlled not-gate (c n−1 not) in various coupling topologies of n qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and swap gates, while the c n−1 not-gate for completely coupled network of six qubits is nearly seven times faster.

Searching for quantum optimal controls under severe constraints

“…Several OCT studies have shown that violating condition (3) by limiting the number of control variables [180] or the control period T [81,168,[170][171][172][173][174][175][176][177][178][179] can prevent the achievement of a globally optimal solution. In this section, we investigate the practical effects of imposing various types of constraints.…”

“…Special algorithms that facilitate successful optimization when the control field has significant spectral constraints have been introduced, for problems such as population transfer in a onedimensional asymmetric double well [169] and molecular alignment [166]. Other works have explored the effect of a specific constraint on OCT optimization; timeoptimal control, the problem of achieving a target objective in the minimum possible time, has received the greatest attention [81,168,[170][171][172][173][174][175][176][177][178][179], and constraints on the number of field components have also been investigated [180]. In this work, we perform extensive OCT simulations to evaluate constraints whose effects on the success of gradient optimization have not previously been examined, identifying values of each constrained parameter beyond which some or all of a set of searches fail to optimize.…”

Time-optimal synthesis of unitary transformations in a coupled fast and slow qubit system

“…[4] (see also [5,6,7,8,9,10,11,12,13,14,15,16,17,18]) presents time-optimal control algorithms to synthesize arbitrary unitary transformations on a system of two qubits. Further progress in the case of multiple qubits is reported in [5,10,16,18,19,20,21,22,23,24,25].…”

“…Given any unitary transformation G ∈ G, let t opt be the smallest possible time such that (a 1 , a 2 ) T ≺ r t opt (b 1 , b 2 ) T (22) and G =K 2 exp[a 1 (−iS β I x )+ a 2 (−iS α I x )]K 1 withK j ∈ K. Again, the KAK decomposition is not unique, and different KAK decompositions correspond to all values a ′ j = a j + 2πz j , where z j ∈ Z (see Sec. A 2).…”

Efficient generation of random multipartite entangled states using time-optimal unitary operations

“…the number of gates needed to reach the Haar distribution. In contraposition to the gate complexity, a new complexity concept for quantum algorithms has been proposed: the time complexity [19,20], understood as the physical time needed to perform such algorithm. The minimization of this time is as important, from the experimental point of view, as the gate complexity.…”