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arXiv:1901.06144 · Journal of Statistical Mechanics Theory and Experiment

Quantum gates by resonantly driving many-body eigenstates, with a focus on Polychronakos’ model

Koen Groenland, Kareljan Schoutens

Published Jul 1, 2019• 4 citations
arXivHTMLPDF

Abstract

Abstract Accurate, nontrivial quantum operations on many qubits are experimentally challenging. As opposed to the standard approach of compiling larger unitaries into sequences of 2-qubit gates, we propose a protocol on Hamiltonian control fields which implements highly selective multi-qubit gates in a strongly-coupled many-body quantum system. We exploit the selectiveness of resonant driving to exchange only 2 out of 2 N eigenstates of some background Hamiltonian, and discuss a basis transformation, the eigengate, that makes this operation relevant to the computational basis. The latter has a second use as a Hahn echo which undoes the dynamical phases due to the background Hamiltonian. We find that the error of such protocols scales favourably with the gate time as t −2 , but the protocol becomes inefficient with a growing number of qubits N . The framework is numerically tested in the context of a spin chain model first described by Polychronakos, for which we show that an earlier solution method naturally gives rise to an eigengate. Our techniques could be of independent interest for the theory of driven many-body systems.

DOI
10.1088/1742-5468/ab25e2

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Abstract Accurate, nontrivial quantum operations on many qubits are experimentally challenging. As opposed to the standard approach of compiling larger unitaries into sequences of 2-qubit gates, we propose a protocol on Hamiltonian control fields which implements highly selective multi-qubit gates in a strongly-coupled many-body quantum system. We exploit the selectiveness of resonant driving to exchange only 2 out of 2 N eigenstates of some background Hamiltonian, and discuss a basis transformation, the eigengate, that makes this operation relevant to the computational basis. The latter has a second use as a Hahn echo which undoes the dynamical phases due to the background Hamiltonian. We find that the error of such protocols scales favourably with the gate time as t −2 , but the protocol becomes inefficient with a growing number of qubits N . The framework is numerically tested in the context of a spin chain model first described by Polychronakos, for which we show that an earlier solution method naturally gives rise to an eigengate. Our techniques could be of independent interest for the theory of driven many-body systems.
We propose a strategy for engineering multiqubit quantum gates. As a first step, it employs an eigengate to map states in the computational basis to eigenstates of a suitable many-body Hamiltonian. The second step employs resonant driving to enforce a transition between a single pair of eigenstates, leaving all others unchanged. The procedure is completed by mapping back to the computational basis. We demonstrate the strategy for the case of a linear array with an even number NNN of qubits, with specific XX+YYXX+YYXX+YY couplings between nearest neighbors. For this so-called Krawtchouk chain, a two-body driving term leads to the iSWAPN{\mathrm{iSWAP}}_{N}iSWAPN​ gate, which we numerically test for N=4N=4N=4 and 6.
We propose a protocol for implementing Barenco-type multi-qubit controlled gates using short driven spin chains. Starting from an Ising interaction with a transverse drive on the last spin, we construct an effective two-qubit Hamiltonian whose time evolution implements the Barenco gate V2(φ,ω,φ)V_2(\varphi,ω,φ)V2​(φ,ω,φ) and, in particular, a CNOT gate. We then embed this construction into a three-qubit XXZXXZXXZ chain to realize the three-qubit Barenco gate V3(φ,ω,φ)V_3(\varphi,ω,φ)V3​(φ,ω,φ), which includes the Toffoli gate as a special case. The derivation is fully analytical: we perform a sequence of unitary transformations, identify decoupled subspaces, and apply a rotating-wave approximation to obtain simple effective Hamiltonians. We derive explicit conditions on the coupling strengths and driving parameters, provide closed-form expressions for the time-evolution operators in each relevant subspace, and characterize the quality of the implementation using the operator fidelity. Numerical simulations show that the protocol achieves high fidelities over broad parameter ranges, demonstrating its robustness and suitability for quantum information processing in spin-chain platforms.
The family of nnn-bit Toffoli gates, with the two-bit Toffoli gate as the figurehead, are of great interest in quantum information as they can be used as universal gates and in quantum error correction, among other things. We present a single-step implementation of arbitrary nnn-bit Toffoli gates (up to a local change of basis), based on resonantly driving a single qubit that has a strong Ising coupling to nnn other qubits. The setup in the two-qubit case turns out to be identical to the universal Barenco gate. The gate time and error are, in theory, independent of the number of control qubits, scaling better than conventional circuit decompositions. We note that our assumptions, namely, strongly coupling n+1n+1n+1 qubits and a driving frequency that scales with nnn, may break down for large systems. Still, our protocol could enhance the capabilities of intermediate-scale quantum computers, and we discuss the prospects of implementing our protocol on trapped ions, Rydberg atoms, and superconducting circuits. Simulations of the latter platform show that the Toffoli gate with two control bits attains fidelities of above 0.98 even in the presence of decoherence. We also show how similar ideas can be used to make a series of controlled-not gates in a single step. We show how these can speed up the implementation of quantum error correcting codes and we simulate the encoding steps of the three-qubit bit-flip code and the seven-qubit Steane code.
The scaling of effort to achieve control objectives with system size is an important consideration in the development of many quantum devices, especially in those intended for quantum information processing. This thesis investigates the scaling with system size of effort required to control manybody quantum systems, and asks whether this is sufficiently favourable to enable supremacy in quantum computation. This is tested using quantum control theory and numerical control optimisation, which are well established tools for investigating the dynamics of driven quantum systems, in a variety of theoretical models related to state-of-the-art quantum devices. Gates acting on a chain of coupled oscillators are found to achievable in times that scale approximately linear with chain length, where the control is over only a single oscillator. The scalability of a scheme for implementing quantum gates in a many-body quantum simulator is shown to be favourable enough to allow for them to be configured to perform quantum computations. A state measurement protocol is proposed as part of the scheme. The scalability is validated through numerical simulation of simulator models composed of up to 9 qubits. The reachable set of operations for systems with quadratic Hamiltonians and infinite-dimensional Hilbert spaces is investigated. The passive operations in quantum optics are proven to be unreachable for single-mode systems with ‘unstable’ Hamiltonians. Further characterisation of the reachable set is made through numerical simulation of control optimisation. Some progress is made in extending the reachability result to n-modes. A non-Markovian noise model is used in simulating control optimisation of a dynamical decoupling scheme, which would be impossible to simulate with Markovian dynamics. The popular QuTiP Python library has been extended to allow simulation of control optimisation with a range of dynamical models. A description of these software modules and their method of use is given. Many of the software tools developed for the study are made available through open-source repositories. Some outlook is given for the use of in-situ open-loop control in optimising controls in quantum system experiments.
We consider the problem of selectively controlling couplings in a practical quantum processor with always-on interactions that are diagonal in the computational basis, using sequences of local not gates. This methodology is well known in nuclear magnetic resonance implementations, but previous approaches do not scale efficiently for the general fully connected Hamiltonian, where the complexity of finding time-optimal solutions makes them only practical up to a few tens of qubits. Given the rapid growth in the number of qubits in cutting-edge quantum processors, it is of interest to investigate the applicability of this control scheme to much larger-scale systems with realistic restrictions on connectivity. Here we present an efficient scheme to find near time-optimal solutions that can be applied to engineered qubit arrays with local connectivity for any number of qubits, indicating the potential for practical quantum computing in such systems.
Single-qubit quantum gates are implemented as the time-controlled unitary evolution of a two-level system under harmonic driving. The rotating-wave approximation (RWA) provides closed-form gate parameters for weak coupling. Larger amplitudes, desirable to speed up the dynamics, make the RWA-designed gates inaccurate because of the neglected counter-rotating term, which also shifts the resonance (Bloch-Siegert shift). Here we study theoretically the infidelity of gates beyond the RWA, using the effective Magnus-Taylor Hamiltonian through second order in the inverse of the driving frequency. By detuning the driving frequency by an amplitude-dependent shift of Bloch-Siegert type and rescaling the operation time, the RWA gate conditions are recovered with corrected parameters. Numerical solutions of the Schrodinger equation show that these corrections lower the gate infidelity by several orders of magnitude, extending the range of usable amplitudes