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Testing quantum fault tolerance on small systems D.

Willsch,1,

2 M. Willsch,1,

2 F. Jin,1 H. De Raedt,3 and K. Michielsen1,

2 1 Institute for Advanced Simulation, J¨ ulich Supercomputing Centre, Forschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany

2 RWTH Aachen University, D-52056 Aachen, Germany

3 Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands (Dated: November 29, 2018) We extensively test a recent protocol to demonstrate quantum fault tolerance on three systems: (1) a real-time simulation of ?ve spin qubits coupled to an environment with two-level defects, (2) a real-time simulation of transmon quantum computers, and (3) the 16-qubit processor of the IBM Q Experience. In the simulations, the dynamics of the full system is obtained by numerically solving the time-dependent Schr¨ odinger equation. We ?nd that the fault-tolerant scheme provides a systematic way to improve the results when the errors are dominated by the inherent control and measurement errors present in transmon systems. However, the scheme fails to satisfy the criterion for fault tolerance when decoherence e?ects are dominant. Keywords: quantum computation;

quantum circuits;

quantum error correction;

quantum information;

fault- tolerance thresholds I. INTRODUCTION A functional universal gate-based quantum computer requires a very high level of precision in implementing the quantum gates. In particular when the devices become bigger, it proves di?cult to maintain this high level of qubit control [1C5] or to satisfy the requirements needed for a computing device [6]. To overcome these limita- tions, the most prominent solution is provided by the theory of fault-tolerant quantum computation [7C9]. However, despite many experiments on quantum codes [10C14], it has still remained an open question how much a practical application can pro?t from a full fault-tolerant protocol. Therefore, Gottesman proposed a test [15] that uses four physical qubits to encode two logical qubits, in combination with a criterion for a successful demonstra- tion of fault tolerance, requiring that All encoded circuits of some representative set perform better than the corresponding bare, unencoded circuits. The underlying error-detecting four-qubit code [16C18] has been implemented with ion-trap qubits [19] and on IBM'

s ?ve-qubit processor [20C22]. Each of these experi- ments reports a successful result, but none explicitly tests the proposed fault-tolerance criterion. In this paper, we report on an extensive test of the fault-tolerance criterion for three complementary sys- tems. System (1) consists of ?ve spin qubits coupled to an environment at a given temperature. We consider various weak- and strong-coupling strengths and various temperatures. This system serves as a general model to study decoherence [23C25]. System (2) is an upscaled ver- sion of the real-time circuit-Hamiltonian simulation used in [5] comprising ?ve transmons and six resonators. Sys- tem (3) is the physical 16-qubit device ibmqx5 provided by IBM [4]. We ?nd very good agreement between the latter two systems for the proper set of optimized gate pulses including measurement errors. The real-time dynamics of both system (1) and (2) are studied by numerically solving the time-dependent Schr¨ odinger equation (TDSE) with = 1, i ? ?t |Ψ(t) = H(t) |Ψ(t) , (1) where H(t) is the time-dependent model Hamiltonian and |Ψ(t) represents the state of the device at time t. Note that the computer simulation is a deterministic pro- gram that always produces the same mathematical solu- tion |Ψ(t) , from which we can compute any physically relevant quantity (such as reduced density matrices of smaller subsystems with non-unitary dynamics) without the need of sampling events. A simulation at this level goes, by de?nition, beyond perturbative studies, mas- ter equations, and assumed Markovianity or completely- positive trace-preserving maps [26C28]. We ?nd that, despite the goal of quantum error cor- rection, the fault-tolerant scheme fails to satisfy the suc- cess criterion under the in?uence of decoherence errors in system (1). However, our study suggests that fault- tolerant schemes can systematically improve the perfor- mance with respect to the natural control and measure- ment errors dominating the transmon systems (2) and (3). This paper is structured as follows. In Sec. II, we give a brief overview of the theory of quantum fault tolerance and the protocol that we study. Section III contains the results for system (1). In this system, there are no control errors, allowing us to assess the performance of the fault- tolerant protocol in the presence of decoherence errors only. In Sec. IV, we present the transmon simulation model, i.e. system (2). This system allows us to study the protocol'