The task of designing control pulses for a system which has not been completely char- acterised can also be viewed as a learning problem in a computer science sense. Machine learning as a classical field has had huge success in recent years due to the develop- ment of deep neural networks [LBH15], and is also being applied to quantum problems [PWZVS16, BWPRW17]. Such approaches have been used in a control context in two different ways: firstly by using classical networks to address quantum control problems, and secondly by making fully quantum networks. In both cases, techniques from machine learning are used to learn how to reach target operations more efficiently through a series of measurements and optimisation. These methods are used in [WDKGK17] to perform small scale quantum communication tasks, which also includes a proposal to scale this up using quantum neurones. Our in situ approach is focussed on implementing gates (and error correcting codes in§VI) without the machinery of neural networks, but share many of the steps in measurement and gradient based optimisation, and has been demonstrated numerically on a larger scale.
That there has been a considerable amount of work in the last three years in this area suggests that it is increasingly seen as a powerful tool in developing good control pulses. A key difference between this prior work and what we present is that the aim here is to discover a control sequence that implements a gate on a large quantum system. The focus is strongly on all aspects of the scaling relation with the number of qubits. Equally importantly, the results are used for a realistic near future quantum simulator and show that it is a viable proposal for turning it into a computer.
V.B
The Scheme
The model considered here is a quantum simulator consisting of n qubits with some in- teractions between them such that they form a fully connected graph. Furthermore it is required that the timescale associated with this interaction is much shorter than the deco- herence time in order for significant entanglement to be built up. In addition to this the ability to perform the following operations on each qubit individually is needed: prepara- tion in a complete basis set of states, fast rotations by applying strong Hamiltonians, and measurement in a complete basis set. These requirements are significant, but much easier than demanding direct control over two-qubit operations, and can be achieved (or is close to being achieved) for the platforms described in§V.A.2.
Those requirements are more than enough for it to be, in principle, possible to do an arbitrary gate on a quantum simulator [Llo95, DNBT02, BBBG09], and to do process tomography [PCZ97]. While other systems satisfy these conditions and the approach detailed here would work, the focus is on this model for clarity. While it is easy to know that a given quantum simulator can do a universal set of gates (by using Lie algebraic methods as described in§I.C.2), finding the time-dependency for the control Hamiltonians
CHAPTER V. IN SITU UPGRADE OF QUANTUM SIMULATORS TO COMPUTERS Initial Control Parameters Propagate Evaluate Fidelity Efficiently Update Control Parameters Target Reached
Figure V.2 |Outline of the process used in optimal control in situ, these are identical to Fig.I.3, except that the steps in red are done on the quantum simulator itself. The propagation is done completely in situ, while the fidelity estimation requires some classical processing too.
that give the desired gate is substantially harder.
The steps for finding such controls for a gate in a purely classical scheme were described in §I.C.3; the key differences between that approach and an in situ one are outlined in Fig.V.2. The starting point in both cases is an initial set of controls that parametrise the strength of the control Hamiltonians over the gate duration, in our examples these are generated randomly. The evolution of the system with these parameters is then cal- culated. On a classical computer this requires solving the time-dependent Schr¨odinger equation numerically for a model of the system, while in this scheme this is simply imple- menting the controls on the simulator. Evaluating the gate fidelity in the classical case is straightforward but, when done in situ, requires some form of tomography to measure it. Our derivation that a tight bound for this gate fidelity can be measured efficiently is given in§V.C. If this fidelity is above a threshold, the process terminates successfully, otherwise the control parameters are updated based on the results of the latest and previous runs, and the process repeats. There are many ways to update the controls classically that can also be used in this method. In the examples described in§V.D a steepest ascent method was used which requires the gradient of the fidelity in the control parameters to also be measured; this can also be done efficiently. The difficulty with doing these steps classically is in computing what unitary is produced by a given choice of control parameters; this requires both an accurate model of a high dimensional system and an exponentially large classical computer to solve it. Neither of those things can be done for a quantum system large enough to be an interesting quantum computer capable of doing things that said classical computer cannot.
These twin difficulties are eliminated by using the quantum simulator to compute the effects of the control pulse on itself. This works because the simulator with a trial set of controls is, evidently, guaranteed to be an accurate model of itself with those controls. The propagation step is therefore done in situ, but the method by which the control parameters are updated remains purely classical. This is because the information extracted from the