Quantum gate learning in qubit networks: Toffoli gate without time-dependent control

Abstract

We put forward a strategy to encode a quantum operation into the unmodulated dynamics of a quantum network without the need for external control pulses, measurements or active feedback. Our optimisation scheme, inspired by supervised machine learning, consists in engineering the pairwise couplings between the network qubits so that the target quantum operation is encoded in the natural reduced dynamics of a network section. The efficacy of the proposed scheme is demonstrated by the finding of uncontrolled four-qubit networks that implement either the Toffoli gate, the Fredkin gate or remote logic operations. The proposed Toffoli gate is stable against imperfections, has a high fidelity for fault-tolerant quantum computation and is fast, being based on the non-equilibrium dynamics. INTRODUCTIONComputational devices based on the laws of quantum mechanics hold promise to speed up many algorithms known to be hard for classical computers. 1 The implementation of a full-scale computation with existing technology requires an outstanding ability to maintain quantum coherence (i.e., isolation from the environment) without compromising the ability to control the interactions among the qubits in a scalable way. Among the most successful paradigms of quantum computation, there is the 'circuit model', in which the algorithm is decomposed into an universal set of single-and two-qubit gates, 2 and, to some extent, the so-called adiabatic quantum computation (AQC), 3 in which the output of the algorithm is encoded in the ground state of an interacting many-qubit Hamiltonian. A different approach 4 is based on the use of always-on interactions, naturally occurring between physical qubits, to accomplish the computation. Compared with the circuit model, this scheme has the advantage of requiring minimal external control and avoiding the continuous switch off and on of the interactions between all but two qubits, whereas compared with AQC it has the advantage of being faster, being based on the non-equilibrium evolution of the system. Quantum computation with always-on interactions is accomplished by combining the natural couplings with a moderate external control, e.g., with a smooth shifting of Zeeman energies, 5 via feed-forward techniques, 6 using measurement-based computation 7 or quantum control. 8,9 Most of these approaches are based on the assumption that the natural couplings are fixed by nature and not tunable, whereas local interactions can be modulated with external fields. However, the amount of external control required can be minimised if the couplings between the qubits can be statically tuned 10 -e.g., during the creation of the quantum device.The recent advances in the fabrication of superconducting quantum devices has opened up to the realisation of interacting quantum networks. In a superconducting device, the qubits are built with a Josephson tunnel element, an inductance and a

Quantum arithmetics via computation with minimized external control: The half-adder

“…Indeed, in Ref. [7] it was shown that both the Toffoli gate and Fredkin gate can be carried out with high fidelity by unmodulated interactions, even when one is restricted to physical two-body couplings between qubits. The procedure for learning the correct physical Hamiltonian that reproduces the given target gate was called quantum gate learning [7,37], while the resulting architecture was called while-you-wait computing [7].…”

“…For this reason, SGD was also used in [7] for obtaining a quantum network that accurately approximates a Toffoli gate. SGD has some benefits: it is fast and has the ability to escape, in principle, from some local minima.…”

“…While-you-wait (WYW) computing has been proposed as a method to combine elements of digital and analog computation [7]. As in analog computing, some computation is implemented into the physical evolution (here unitary dynamics) of an interacting many-particle system.…”

“…A fruitful approach is to view the many-body system as a quantum neural network that has to "learn" a desired quantum transformation [7,[28][29][30][31], or, alternatively, to use powerful optimization algorithms [32][33][34][35]. Here we use differential evolution [36], a global optimization algorithm inspired by evolutional biology, which has been recently successfully applied in quantum control problems [34].…”

“…The transmission can be either obtained via optical means, or with one-dimensional (1D) qubit chains, to avoid the hybrid setups. Combining WYW computing with low-control methods to transfer states in 1D chains [8][9][10][11][12][13][14][15][16] would allow a minimal control approach to computation [7].…”