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.

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This paper is dedicated to the memory of Martti Salomaa.Quantum optimal control theory is applied to two and three coupled Josephson charge qubits. It is shown that by using shaped pulses a cnot gate can be obtained with a trace fidelity > 0.99999 for the two qubits, and even when including higher charge states, the leakage is below 1%. Yet, the required time is only a fifth of the pioneering experiment [1] for otherwise identical parameters. The controls have palindromic smooth time courses representable by superpositions of a few harmonics. We outline schemes to generate these shaped pulses such as simple network synthesis. The approach is easy to generalise to larger systems as shown by a fast realisation of Toffoli's gate in three linearly coupled charge qubits. Thus it is to be anticipated that this method will find wide application in coherent quantum control of systems with finite degrees of freedom whose dynamics are Liealgebraically closed.PACS numbers: 85.25. Cp, 82.65.Jn, 03.67.Lx, 85.35.Gv In view of Hamiltonian simulation and quantum computation recent years have seen an increasing amount of quantum systems that can be coherently controlled. Next to natural microscopic quantum systems, a particular attractive candidate for scalable setups are superconducting devices based on Josephson junctions [2]. Due to the ubiquitous bath degrees of freedom in the solid-state environment, the time over which quantum coherence can be maintained remains limited, although significant progress has been achieved [3,4]. Yet, it is a challenge how to produce accurate quantum gates, and how to minimize their duration such that the number of possible operations within T 2 meets the error correction threshold. Concomitantly, progress has been made in applying optimal control techniques to steer quantum systems [5] in a robust, relaxation-minimising [6] or timeoptimal way [7]. Spin systems are a particularly powerful paradigm of quantum systems [8]: under mild conditions they are fully controllable, i.e., local and universal quantum gates can be implemented. In N spins-1 2 it suffices that (i) all spins can be addressed selectively by rf-pulses and (ii) that the spins form an arbitrary connected graph of weak coupling interactions. The optimal control techniques of spin systems can be extended to pseudo-spin systems, such as charge or flux states in superconducting setups, provided their Hamiltonian dynamics can be approximated to sufficient accuracy by a closed Lie algebra, e.g., in a system of N qubits su(2 N ).As a practically relevant and illustrative example, we consider two capacitively coupled charge qubits controlled by DC pulses as in Ref. [1]. The infinitedimensional Hilbert space of charge states in the device can be projected to its low-energy part defined by zero or one excess charge on the respective islands [2]. Identifying these charges as pseudo-spin states, the Hamiltonian can be written as H tot = H drift + H control , where the drift or static part reads (for the constants see caption to Fig. 1)while...

Optimal control methods for implementing quantum modules with least amount of relaxative loss are devised to give best approximations to unitary gates under relaxation. The potential gain by optimal control fully exploiting known relaxation parameters against time-optimal control (the alternative for unknown relaxation parameters) is explored and exemplied in numerical and in algebraic terms: for instance, relaxation-based optimal control is the method of choice to govern quantum systems within subspaces of weak relaxation whenever the drift Hamiltonian would otherwise drive the system through fast decaying modes. In a standard model system generalising ideal decoherence-free subspaces to more realistic scenarios, opengrape-derived controls realise a cnot with delities beyond 95% instead of at most 15% for a standard Trotter expansion. As additional benet their control elds are orders of magnitude lower in power than bang-bang decouplings.

The repertoire of problems theoretically solvable by a quantum computer recently expanded to include the approximate evaluation of knot invariants, specifically the Jones polynomial. The experimental implementation of this evaluation, however, involves many known experimental challenges. Here we present experimental results for small-scale approximate evaluation of the Jones polynomial by nuclear magnetic resonance (NMR); in addition, we show how to escape from the limitations of NMR approaches that employ pseudopure states. Specifically, we use two spin-1/2 nuclei of natural abundance chloroform and apply a sequence of unitary transforms representing the trefoil knot, the figure-eight knot, and the Borromean rings. After measuring the nuclear spin state of the molecule in each case, we are able to estimate the value of the Jones polynomial for each of the knots.

The definition of a cluster state naturally suggests an implementation scheme: find a physical system with an Ising coupling topology identical to that of the target state and evolve freely for a time of 1 2J . Using the tools of optimal control theory, we address the question of whether or not this implementation is time-optimal. We present some examples where it is not and provide an explanation in terms of geodesics on the Bloch sphere.

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Quantum control plays a key role in quantum technology, in particular for steering quantum systems. As problem size grows exponentially with the system size, it is necessary to deal with fast numerical algorithms and implementations. We improved an existing code for quantum control concerning two linear algebra tasks: The computation of the matrix exponential and efficient parallelisation of prefix matrix multiplication.For the matrix exponential we compare three methods: the eigendecomposition method, the Padé method and a polynomial expansion based on Chebyshev polynomials. We show that the Chebyshev method outperforms the other methods both in terms of computation time and accuracy. For the prefix problem we compare the tree-based parallel prefix scheme, which is based on a recursive approach, with a sequential multiplication scheme where only the individual matrix multiplications are parallelised. We show that this fine-grain approach outperforms the parallel prefix scheme by a factor of 2-3, depending on parallel hardware and problem size, and also leads to lesser memory requirements.Overall, the improved linear algebra implementations not only led to a considerable runtime reduction, but also allowed us to tackle problems of larger size on the same parallel compute cluster.