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Quantifying the Quantum Gate Fidelity of Single-Atom Spin Qubits in Silicon

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Quantifying the Quantum Gate Fidelity of Single-Atom Spin Qubits in Silicon Quantifying the quantum gate fidelity of single-atom spin qubits in silicon by randomized benchmarking J T Muhonen,1, ∗ A Laucht,1 S Simmons,1 J P Dehollain,1 R Kalra,1 F E Hudson,1 S Freer,1 K M Itoh,2 D N Jamieson,3 J C McCallum,3 A S Dzurak,1 and A Morello1, y 1Centre for Quantum Computation and Communication Technology, School of Electrical Engineering and Telecommunications, UNSW Australia, Sydney, New South Wales 2052, Australia 2School of Fundamental Science and Technology, Keio University, 3-14-1 Hiyoshi, 223-8522, Japan 3Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Melbourne, Victoria 3010, Australia Building upon the demonstration of coherent control and single-shot readout of the electron and nuclear spins of individual 31P atoms in silicon, we present here a systematic experimental estimate of quantum gate fidelities using randomized benchmarking of 1-qubit gates in the Clifford group. We apply this analysis to the electron and the ionized 31P nucleus of a single P donor in isotopically purified 28Si. We find average gate fidelities of 99.95% for the electron, and 99.99% for the nuclear spin. These values are above certain error correction thresholds, and demonstrate the potential of donor-based quantum computing in silicon. By studying the influence of the shape and power of the control pulses, we find evidence that the present limitation to the gate fidelity is mostly related to the external hardware, and not the intrinsic behaviour of the qubit. I. INTRODUCTION number of qubits since the number of measurements re- quired to completely map a multi-qubit process increases The discovery of quantum error correction is among exponentially with the number of qubits involved. the most important landmarks in quantum information In addition, and more importantly for the single qubit science [1{4]. Together with the steady improvement in system discussed here, quantum process tomography is the coherence and fidelity of various physical quantum also sensitive to errors in the state preparation and mea- bit (qubit) platforms, it represents one of the main moti- surement (SPAM errors) and cannot distinguish between vations for the investment in quantum information tech- these and pure control errors [12]. Therefore, with QPT nologies. In general, uncontrolled interactions between it is only possible to characterize gate fidelities if the gate the qubit and its environment can cause loss of coher- errors are larger than preparation and readout errors. ence and gate errors. Quantum error correcting codes, This is an unsatisfying situation, since fault-tolerant however, guarantee that the errors can be recovered, quantum computation normally poses much more strin- provided that the average error rate is below a certain gent requirements on gate fidelities than on initialization fault-tolerance threshold [5]. The threshold is strongly and readout. dependent on the quantum computer architecture: for Randomized benchmarking [13, 14] has gained atten- instance, a bilinear nearest-neighbour array requires er- tion in recent years as a scalable solution for determin- rors below 10−6 to achieve fault-tolerance [6]. More re- ing gate fidelities, and has been experimentally demon- cent ideas have shown the tantalizing prospect of fault- strated with e.g. atomic ions [15, 16], nuclear magnetic tolerance error correction with error rates that can be as resonance [17] and superconducting qubits [9, 12]. The high as 10−2 [7, 8]. This level of gate fidelity has become goal of randomized benchmarking protocols is to extract accessible to the most advanced qubit platforms, such as the average gate fidelity, defined as the average fidelity superconducting qubits [9] and trapped ions [10]. of the output state over pure input states. The fidelity Quantifying gate errors is, however, not trivial. The of the output state E(ρ) as compared to the ideal output most commonly used protocol, quantum process tomog- state U(ρ) is defined as [3] raphy (QPT) [11], is based upon preparing different input 2 states (chosen to form a complete basis) and processing qp p arXiv:1410.2338v1 [quant-ph] 9 Oct 2014 F = tr E(ρ)U(ρ) E(ρ) ; (1) them identically many times. The output states are then U characterized with quantum state tomography to extract all the components of the process matrix and quantify where U is the ideal gate operation, E the actual oper- the process errors. QPT can in principle be applied to ation and ρ is the density matrix describing the input any process and state space, but it is not scalable to large state. Here we adopt a benchmarking method based upon the construction and application of random sequences of gates that belong to the Clifford group. It has been ∗ [email protected] shown [18{20] that the average fidelity of a Clifford gate y [email protected] can be extracted from the fidelity of the final state of the 2 qubit, averaged over several random sequences of Clifford is nuclear gyromagnetic ratio [29]. The experiments were gates, having started from a fixed input state. This pro- performed in high magnetic fields (B0 ≈ 1:5 T applied vides a scalable benchmarking method with well-defined along the [110] crystal axis of Si) and low temperatures conditions of applicability, and which does not depend (electron temperature Tel ≈ 100 mK). Microwave control on SPAM errors [18{20]. The Clifford group does not fields B1 are produced by a broadband on-chip microwave provide a universal set of quantum gates, but a univer- antenna [30] terminating ∼ 100 nm away from the donor sal set can be constructed from them by the addition of qubit. At microwave frequencies, the cable and the cold only one gate or, alternatively, by using ancilla qubits attenuators connecting the source to the on-chip antenna and their measurements [3]. Also, the Clifford gates play provide ∼ 40 dB attenuation. an important role in the error correction schemes based The device structure is shown in figure 1(a). The sub- on stabilizer codes [21]. strate is a 0.9 µm thick epilayer of isotopically purified Spin qubits based upon the electron or nuclear spin 28Si, grown on top of a 500 µm thick natSi wafer (figure of phosphorus atoms in isotopically purified 28Si [22] are 1(b)). The 28Si epilayer contains 800 ppm residual 29Si known to have extraordinary coherence times [23, 24], isotopes. A stack of aluminium gates above the SiO2 is and recent experiments have shown that such record co- used to induce a single-electron transistor (SET) under- herence can be retained also at the single-atom level in neath the oxide. The single-donor qubit is selected out of functional nanoelectronic devices [25]. Here, we present a a small number of ion-implanted 31P atoms [31], placed thorough investigation of the gate fidelities of the single- ≈ 10 nm below the Si/SiO2 interface and underneath an atom spin qubit device described in Ref. [25]. We focus additional stack of control gates. The single-shot elec- on the gate fidelity of qubits represented by the elec- tron spin readout is based on its spin-dependent tunnel- tron (e−) and ionized nuclear spin (31P+) with random- ing [27, 32] into the nearby SET island. The readout ized benchmarking protocols using Clifford-group gates. process also leaves the electron spin initialized in the j#i We show that all 1-qubit gate fidelities are consistently state. Since this only occurs when the donor is brought above 99.8 % for the electron (with a deduced average in resonance with the Fermi level of the detector SET, of 99.95 % from a long sequence of Clifford gates) and we can tune the electrostatic gates fabricated above the above 99.9 % for the nucleus (with an average of 99.99 %). donor implant area to ensure that, at any time, only a These results, combined with the previously reported single donor is able to undergo electron tunneling events. record coherence times, show that individual dopants in The successful isolation of a single 31P donor is unequiv- 28Si are one of the best physical realizations of quantum ocally proven by ESR experiments where only one of the bits. ESR frequencies is active (figure 1(c)) at any time. The ESR frequency sporadically jumps between νe1 and νe2, signalling the flip of the nuclear spin state [29]. The nu- II. SINGLE-ATOM SPIN QUBIT DEVICE clear spin readout is obtained by applying a π-pulse at frequency νe2 to an electron initially j#i. If the electron is successfully flipped to the j"i state, the nuclear spin Our qubit system is a single substitutional 31P donor state is declared j*i. in silicon, fabricated and operated as described in detail in [25{29]. In particular, the system used in the present experiment is the same as Device B in [25]. Both the donor-bound electron (e−) and the nucleus (31P) possess III. RANDOMIZED BENCHMARKING a spin 1=2 and each encode a single qubit. The qubit logic PROTOCOL FOR AVERAGE CLIFFORD GATE states are the simple spin up/down eigenstates, which we FIDELITY denote with j"i, j#i for the electron, and j*i, j+i for the nucleus. The electron and the nucleus are coupled by The randomized benchmarking protocol we adopt is the contact hyperfine interaction A. In the presence of a based upon the one studied theoretically in [18{20] and large magnetic field B0, the resulting eigenstates are, to a demonstrated experimentally with e.g. superconducting very good approximation, the separable tensor products qubits and silicon quantum dots in [9, 33]. It is based on of the electro-nuclear basis states (j*"i, j*#i, j+"i, j+#i).
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