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Fast Noise-Resistant Control of Donor Nuclear Spin Qubits in Silicon

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Fast Noise-Resistant Control of Donor Nuclear Spin Qubits in Silicon Fast noise-resistant control of donor nuclear spin qubits in silicon James Simon1;2, F. A. Calderon-Vargas2, Edwin Barnes2, and Sophia E. Economou2∗ 1Department of Physics, University of California, Berkeley, California 94720, USA 2Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA A high degree of controllability and long coherence time make the nuclear spin of a phosphorus donor in isotopically purified silicon a promising candidate for a quantum bit. However, long- distance two-qubit coupling and fast, robust gates remain outstanding challenges for these systems. Here, following recent proposals for long-distance coupling via dipole-dipole interactions, we present a simple method to implement fast, high-fidelity arbitrary single- and two-qubit gates in the absence of charge noise. Moreover, we provide a method to make the single-qubit gates robust to moderate levels of charge noise to well within an error bound of 10−3. I. INTRODUCTION on clock transitions. Our approach is based on using optimally designed control pulse waveforms and energy Nuclear spins in the solid state present unparalleled transition modulation. Accordingly, we derive a time- advantages as a platform for quantum computing due independent analytical approximation for the system's to their long coherence times [1, 2] and high degree of Hamiltonian, explain how to rapidly implement arbi- controllability [3{5]. In particular, the nuclear spin of a trary, noise-resistant single-qubit gates with fidelities ex- phosphorus donor in silicon is a promising candidate for a ceeding 99.9% even in the presence of significant charge quantum bit [6, 7] owing to its coherent control [8{10] and noise, and provide a method to use the dipole-dipole in- minute-long coherence time [2]. The use of isotopically- teraction to implement cphase gates across a distance purified silicon nanostructures [11] considerably reduces of 0:5 µm. The single-qubit and cphase gates have max- magnetic environmental noise, allowing high-fidelity con- imum durations of 500 ns and 750 ns, respectively, with trol [8]. However, long coherence times are only use- special cases such as single-qubit Z gates being much ful when gates are very fast in comparison. One of faster (< 25 ns). An advantage of our protocol compared the difficulties of using the nuclear spin as a qubit has to prior work [16] is that it does not require finely tuning been the implementation of fast single- and two-qubit the system to a clock transition. This is not done at the gates. Controlling a nuclear spin with an oscillating expense of gate performance, and our gates are as robust magnetic field as in nuclear magnetic resonance is slow, but faster than those of Ref. 16. with typical gate times ranging from a few to tens of mi- The paper is organized as follows. In Sec. II we in- croseconds [8, 9, 12]. Moreover, most of the approaches troduce the system and its Hamiltonian. In Sec. III, we for multi-qubit operations require short interaction dis- derive an analytical time-independent Hamiltonian fol- tances [6][13], and thus demand near-atomic precision in lowing two approaches. In Sec. IV, we define the qubit the placement of the donors [14, 15]. states and explain how to implement robust single-qubit To overcome these challenges, Ref. 16 proposes the dy- gates, specifically arbitrary Z-rotations and X-rotations. namical creation of a strong electric dipole transition at We give a method for implementing fast controlled-phase microwave frequencies for the nuclear spin by sharing an gates between two adjacent qubits separated by a dis- electron between the donor and the Si/SiO2 interface and tance of 0:5 µm in Sec. V. We conclude in Sec. VI. applying an oscillating magnetic field. This facilitates the implementation of two-qubit gates via dipole-dipole in- teractions or, alternatively, the qubit's coupling to other II. THE SYSTEM quantum systems. Moreover, nuclear spin-flip transitions can also be sped up by including an oscillating electric arXiv:2001.10029v2 [quant-ph] 20 May 2020 field along with the magnetic drive. A potential down- The system follows the experimental proposal reported in Refs. 16 and 17, where a donor 31P atom is embed- side of making the system amenable to electrical control 28 is that it increases its sensitivity to charge noise due to ded in enriched Si a distance d away from a Si=SiO2 the charge component in the states encoding the qubit. interface, as shown in Figure 1. The donor atom pro- vides a nuclear spin I = 1=2 with gyromagnetic ratio However, Ref. 16 shows that there are regions in param- −1 γn=2π = 17:23 MHzT and a free electron with spin eter space (\clock transitions") where the nuclear spin −1 transition is insensitive to electrical noise to at least first S = 1=2 and gyromagnetic ratio γe=2π = 27:97 GHzT . order. The electron and nuclear spins are coupled via a hyper- In this work, we propose an alternative path toward fine interaction with coupling strength A, which is ap- robust high-fidelity single-qubit gates that does not rely proximately equal to 117 MHz when the electron is bound to the nucleus. A metal gate positioned on top of the donor atom is used to control the position of the electron via electric fields, which also tunes the hyperfine inter- ∗ [email protected] action A down to zero when the electron is at the inter- 2 Note also that all-electrical spin control is possible even in the presence of a significant spin-orbit interaction, but it is vulnerable to charge noise [20]. The Hamiltonian of the system, therefore, consists of the orbital part, the Zeeman part, and the hyperfine cou- pling: H = Horb + HB + HA: (1) Here each term can be expressed in terms of the electron id id position operators τz = jii hij − jdi hdj, τx = jii hdj + jdi hij, and the electron (nuclear) spin operator S (I) as follows: de (∆E + Ea cos [!Et]) id Vt id Horb = − τz + τx ; 2~ 2 1 id − τz HB =B0 γe 1 + ∆γ Sz − γnIz 2 (2) 31 FIG. 1. One nuclear spin qubit in the system described. A P + B cos [! t](γ S − γ I ) ; donor is embedded in 28Si, and the free electron in the system a B e x n x can be pulled towards a Si-SiO interface. The electron orbit 1 − τ id 2 H =A z S · I; is quantized into a jdi state on the donor and a jii state on the A 2 interface, and both the electron and nuclear spins are used. Static and oscillating electric and magnetic fields are used where ∆E = E − E0 is the deviation of the electric field to control the system. The qubit is ultimately stored in the away from the ionization point. nuclear spin state, with the other degrees of freedom used for The qubit is encoded in the two lowest-energy eigen- driving gates. states of the system, which, in the absence of AC driv- ing, are approximately jg #*i and jg #+i, where jgi is the ground eigenstate of the orbital part of the Hamiltonian face. Moreover, the gyromagnetic ratio of the electron with no AC fields. Therefore, it is convenient to express bound to the nucleus can differ from that of an electron the total Hamiltonian in (1) in a basis spanned by the at the interface by an amount ∆γ that can reach up to orbital eigenstates fjgi ; jeig. The electron position oper- id id 0.7% [18]. Therefore, the Hilbert space includes three ators τx and τz in the orbital eigenbasis are binary degrees of freedom: the nuclear spin of the donor id de∆E Vt atom, the spin of the free electron, and the position of τ = τz + τx; z " " the free electron, which is quantized into a state on the ~ 0 0 (3) donor atom, jdi, and a state at the interface, jii, which id Vt de∆E τx = − τz + τx; is a good approximation as demonstrated by Ref. 17. "0 ~"0 The system has two control fields, one electric and one where τz = jgi hgj − jei hej and τx = jgi hej + jei hgj are magnetic, each with static (DC) and oscillating (AC) p 2 2 the orbital operators, and "0 = Vt + (de∆E=~) is the components. The DC component of the electric field, orbital (charge) energy splitting. The Hamiltonian com- E, points along the donor-interface axis and controls ponents in Eq. 2 have the following form in the basis the electron position (see Fig. 1), determining the am- spanned by the orbital and spin eigenbasis: plitudes of the states jii and jdi in the orbital ground state. The AC electric field is parallel to E and is given −"0 deEa cos(!Et) de∆E Vt Horb = τz − τz + τx ; by Eac(t) = Ea cos(!Et). In particular, when E = E0, 2 2~ ~"0 "0 where E is the magnitude of the electric field at the ion- 1 0 1 de∆E=~τz + Vtτx ization point, the orbital ground state of the electron has HB =B0γe + − ∆γ Sz 2 2"0 equal probability to be at the donor nucleus and at the − B0γnIz + Ba cos [!Bt](γeSx − γnIx) ; interface. When E E0, the electron is fully on the 1 donor (jdi), and when E E0 the electron is pulled off de∆E Vt HA =A − τz − τx S · I: the donor (jii). The transition frequency between the or- 2 2~"0 2"0 bital ground and excited states at the ionization point is (4) equal to the tunnel coupling Vt.
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