Two-qubit silicon quantum processor with operation fidelity exceeding 99%

Since the publication of the Loss-DiVincenzo proposal for spin-based quantum information processing in 1998 (1), the semiconductor quantum dot research community has worked to satisfy the DiVincenzo criteria for quantum information processing using spin qubits. Electron spins can be initialized and read out using spin-to-charge conversion (2, 3), single-spin qubits can be coherently controlled using oscillating electromagnetic fields (4–6), and nearest-neighbor spins can be coherently coupled via the exchange interaction (3).

Seminal results for spin qubits were obtained with gallium arsenide quantum dots (2–4); however, hyperfine coupling of the electron spin to lattice nuclei greatly limited quantum coherence (7). A transition to silicon, which can be isotopically enriched, has led to a several order-of-magnitude increase in electron spin coherence times (8) and improved quantum control fidelities (9–11). The small ~100-nm scale of quantum dot spin qubits and the substantial capabilities of the silicon microelectronics industry could allow for scaling to large system sizes that are also capable of fault tolerant operation.

While high single- and two-qubit gate fidelities have been demonstrated in silicon (9, 11–13), state preparation and measurement (SPAM) errors have generally hovered around ~10 to 20%. Here, we demonstrate a spin-based two-qubit quantum processor with all-around high-performance fidelities [readout F > 97%, simultaneous single-qubit control F > 99%, and a two-qubit controlled-phase (CPHASE) gate F > 99.8%]. Our two-qubit gate fidelity exceeds recent reports on spin qubits (12, 13) and is competitive with superconducting qubits (14, 15).

High-fidelity quantum control and readout are achieved in the first two qubits (Q1 and Q2) of a six-qubit linear array (Fig. 1A). Quantum dot electrons are vertically confined in an isotopically enriched [800 parts per million (ppm) of residual 29Si] 28Si quantum well, and lateral confinement is achieved using an overlapping aluminum gate stack (16). Figure 1B depicts the double quantum dot formed under gates P1 and P2 with the exchange interaction controlled by gate B2 (3). An external magnetic field BE = 365 mT is applied in the plane of the quantum well to Zeeman-split the spin states. The external field adds to the magnetic field generated by a Co micromagnet, resulting in electron spin resonance frequencies f1 = 18.247 GHz and f2 = 17.851 GHz.

We first demonstrate high-visibility readout and single-qubit control fidelities. Single-qubit gates are achieved at the symmetric operating point in the (1,1) charge state using electric dipole spin resonance (EDSR) in the transverse field gradient created by the micromagnet (6, 17, 18). Here, (N1,N2) denotes the charge occupation of dots 1 and 2. Qubit state preparation and readout are achieved by spin-dependent tunneling with the reservoir, and cryogenic amplifiers are used to improve charge readout fidelity (2, 19, 20, 21). Figure 1C shows Rabi chevrons for each qubit, obtained by measuring the spin-up probability P↑ as a function of drive frequency f and microwave burst length τR. Rabi oscillations approaching unit visibility are achieved when driving each qubit on resonance (Fig. 1D).

We perform the quantum characterization, verification, and validation protocols of gate set tomography (GST) on the single-qubit gates—identity (idle qubit i for π/2 gate time of 70 ns) Ii, πX/2 rotation Xi, and πY/2 rotation Yi—and estimate single-qubit control fidelities above 99.9% when driving and measuring one qubit at a time (Fig. 1C, inset) (22). The fidelities are limited by decoherence [T2*(T2) = 1.7(23) μs and 2.3(102) μs for Q1 and Q2 measured using the Ramsey and Hahn echo pulse sequences] and are comparable to the highest single-spin qubit gate fidelities in the literature (9, 23). The high idling infidelities and large increase in echoed coherence times suggest that the dominant source of decoherence is the low-frequency charge noise coupling to the magnetic field gradient instead of nuclear spin noise. SPAM errors extracted from GST are quantified by the spin-down ground-state initialization fidelities ρ0,1 = 99.4% and ρ0,2 = 97.5% and the measurement fidelities M1 = 98.1% and M2 = 99.8%, making the overall operation fidelity high enough to support common error correction protocols.

Building to the two-qubit space, we use GST to characterize the qubit control fidelities when both qubits are operated simultaneously (X1 ⊗ X2, Y1 ⊗ X2, etc.) by combining microwave control signals on the drive gate (MW gate in Fig. 1A). GST estimates an average simultaneous single-qubit control fidelity F = 99.65 ± 0.07% primarily limited by cross-talk when operating in the two-qubit space (20). Gates that are played sequentially (X1 ⊗ I2, I1 ⊗ X2, etc.) have a lower average fidelity of 98.97 ± 0.2% because of the higher infidelity of idling one qubit while the neighboring qubit is manipulated, so we operate single-qubit gates simultaneously where possible. Last, we cross-check these GST results using the widely accepted protocol of randomized benchmarking (RB) and achieve F1 = 99.50 ± 0.02%, F2 = 99.48 ± 0.02%, and a joint fidelity of 99.13 ± 0.03%. These results build significantly on past attempts to simultaneously drive spin qubits, where single-qubit control fidelities were as low as 97% (10).

Two-qubit control is achieved by pulsing on the barrier gate B2 to turn on the exchange interaction J(VB2) (3). Figure 2A demonstrates a three-decade variation of J(VB2). The exchange interaction is quantified in the high-J regime (J/h > 1/T2*) by measuring time domain exchange oscillations (24), whereas a spin echo is used to measure residual exchange interaction down to a T2 limit of J/h ~ 10 kHz, (blue in Fig. 2A). In the regime where the longitudinal magnetic field gradient exceeds the exchange energy ΔEz >> J, the antiparallel spin states accumulate a spin-dependent phase described by the unitary UCPHASE = diag(1,eiφ2, eiφ1,1), where φ1(2) = J(VB2)twait/2ħ (25, 26). In addition to evolution due to the exchange interaction, each qubit accumulates a phase because of precession in the magnetic field gradient during the barrier pulse. To realize a controlled-Z (CZ) gate UCZ = diag(1,1,1,−1), the barrier is pulsed for twait = πħ/J(VB2) and then Z(ϕ1(2)) gates are applied to each qubit, where ϕ1(2) is the rotation angle that achieves the CZ gate for qubit 1(2). The Z gates here are performed virtually by adding the phase to the subsequent X and Y gates (27).

For these experiments, we first establish the exchange gate time twait and optimize the barrier amplitude J(VB2) using a phase-insensitive optimization routine (20). Then, as shown in Fig. 2B, we optimize the phases for the CZ gate by preparing the target qubit (Q2 here) in a superposition state and the control qubit in either ↑ or ↓. The exchange interaction is then pulsed on for 40 ns using a smoothed square pulse (20) to execute a CPHASE gate. Afterward, a Z(ϕ*) gate is applied to realize a CZ gate. As an initial demonstration of full two-qubit control with low SPAM errors, we perform Bell state tomography (which requires a combination of single- and two-qubit operations) and use maximum likelihood estimation to achieve the Bell state fidelities ∣Ψ+⟩ = 97.5% (98.6%), ∣Ψ−⟩ = 97.0% (98.7%), ∣Φ+⟩ = 95.4% (96.6%), and ∣Φ−⟩ = 96.3% (97.4%), without (with) SPAM corrections included (Fig. 2C) (20). These results significantly improve upon past measurements with SPAM-corrected fidelities of 78 to 90% (11, 24, 28, 29) and are comparable to the simulated Bell state fidelities obtained by Xue et al. (12).

To demonstrate integrated control of the two-qubit processor, we combine the CZ with other primitive qubit operations to create familiar two-qubit gates (e.g., CNOT and SWAP). We first synthesize a CNOT gate using the Hadamard and CZ gates. Figure 3A shows the raw input-output measurement results of performing the synthesized CNOT gate on the four different input product states with Q2 as the target. To show that the target qubit will follow the control qubit state, we prepare Q1 in the ↓ state and Q2 in an arbitrary superposition state using a Rabi drive pulse for time τR. This state preparation routine is followed by the CNOT, with Q1 as the target qubit this time. The result is the high-visibility anticorrelated oscillations of the ↓↓ and ↑↑ joint state probabilities (Fig. 3B). In addition, we generate synthesized SWAP gates with three alternating CNOT operations and measure the input-output SWAP table (Fig. 3C). To show a SWAP of an arbitrary superposition state ∣ψ⟩, we perform a Rabi drive pulse on Q2 and then SWAP the state onto Q1, resulting in Q2 being in the ↓ state and Q1 in the ∣ψ⟩ state (Fig. 3D).

We turn to the two-qubit interleaved RB protocol to quantify the overall performance of our processor (30). The two-qubit Clifford group C2 in this experiment has 576 single-qubit elements and integrates the CZ into 10,944 two-qubit elements containing CNOT-, iSWAP-, and SWAP-like operations similar to those demonstrated in Fig. 3. In this compilation, the average two-qubit Clifford consists of 8.25 single-qubit operations and 1.5 CZ operations (31). Random sequences of these two-qubit Clifford operations are generated, and a unique recovery Clifford that inverts the sequence and returns the qubits to the ↓↓ ground state is appended. The reference sequence fidelity Gref is fit to Gref = AprefM + B, where pref is the sequence decay, M is the number of Cliffords in the sequence, and SPAM errors are absorbed into parameters A and B. The average error per Clifford of the reference is then rref = (1 − pref)(d − 1)/d, with d = 2NQ and with NQ as the number of qubits. With this technique, the CZ and synthesized CNOT fidelities can be determined by interleaving these operations in the benchmarking sequences after every Clifford operation and by comparing to the reference curve. The interleaved gate fidelities, Fgate = 1 − rgate, are extracted using their sequence decay pgate and the relation rgate = (1 − pgate/pref)(d − 1)/d. To thoroughly sample the Clifford group and obtain an accurate estimate of our CZ and CNOT fidelities, we randomize 125 unique sequences for each reference and interleaved measurement going to sequence lengths as long as 65 total Clifford operations and average 160 times (Fig. 4). The resulting two-qubit RB fidelities for the CZ and CNOT are FCZ = 99.81 ± 0.17% and FCNOT = 98.62 ± 0.16%, with error bars determined by bootstrapping (31). The reference error rate is rref = 0.0679 and is consistent with the average composition of the two-qubit Clifford operation rref = 1.5rCZ + 8.25rSQ, CZ error rate rCZ ~0.002, and a joint error rate from simultaneous single-qubit operation rSQ ~0.008.

A natural next step is to extend these operations to three or more qubits and develop optimal protocols for mitigating cross-talk in both the single- and two-qubit operations. In this device, the microwave drive couples nearly equally to both qubits. Cross-talk could be reduced in future experiments by, instead, driving plunger gates, which have low cross-capacitance to neighboring dots. We suspect that charge noise is one of the main limiting factors for the CZ gate, as both Z(ϕ) parameters and exchange amplitude are affected by electrostatic fluctuations in the device. For devices that use engineered magnetic field gradients, reduction of charge noise and improvements to the field gradient profile will be necessary to increase coherence times and to improve gate performance. To advance toward active feedback error correction protocols, the measurement bandwidth will have to be increased.

We have demonstrated full two-qubit control in a silicon quantum device, with simultaneous single-qubit control fidelities exceeding 99% and a primitive two-qubit CZ gate fidelity exceeding 99.8%. In contrast to previous implementations (12, 13, 24, 28), SPAM errors are very low (<3%). Our demonstration represents the highest total operation fidelity in a two-qubit processor realized in silicon quantum dots, with performance capable of fault tolerant operation (32). These experiments demonstrate two-qubit gates with silicon spin qubits at speeds exceeding trapped ions (33) and fidelities comparable with superconducting qubits (14, 15). Given recent advances in quantum dot fabrication (16, 34), spin qubits are poised to scale-up to larger multiqubit quantum processors.

The device is fabricated on a 28Si/SiGe heterostructure with a residual 29Si concentration of 800 ppm in the isotopically purified 5-nm-thick Si quantum well. A multilevel gate stack consisting of three layers of Al gates is used to define the electrostatic confinement potential in the plane of the quantum well. A Co micromagnet provides the field gradient necessary for spin addressability and EDSR.

The measurements are performed in a dilution refrigerator with a base temperature of 12 mK and electron temperature Te ~ 45 mK. Pulses for IQ modulation, exchange gates, and readout are generated using a Tektronix 5208 arbitrary waveform generator. Two Agilent E8267D microwave signal generators provide the EDSR pulses that are applied to the microwave gate shown in Fig. 1A.

We thank R. Delva for computational support and D. Zajac for assistance with sample fabrication. The Si/SiGe heterostructure used in these experiments was provided by HRL Laboratories LLC. Supported by Army Research Office grant W911NF-15-1-0149 and DARPA grant D18AC0025. Devices were fabricated in the Princeton University Quantum Device Nanofabrication Laboratory, which is managed by the Department of Physics. We acknowledge the use of Princeton’s Imaging and Analysis Center, which is partially supported by the Princeton Center for Complex Materials, a National Science Foundation MRSEC program (DMR-2011750). The Sandia National Laboratories part of this work was funded, in part, by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research’s Quantum Testbed Pathfinder. Sandia National Laboratories is a multimission laboratory managed and operated by the National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. All statements of fact, opinion, or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI, the U.S. Department of Energy, or the U.S. government.

Funding: This work was supported by Army Research Office grant W911NF-15-1-0149 (to A.R.M., A.J.S., C.R.G., M.J.G., and J.R.P.), DARPA grant D18AC0025 (to M.J.G., M.M.F., and J.R.P.), National Science Foundation grant DMR-2011750 (to A.R.M. and J.R.P.), and Department of Energy grant DE-NA0003525 (to E.N.).

Author contributions: Conceptualization: A.J.S., A.R.M., and J.R.P. Methodology: A.J.S., A.R.M., C.R.G., E.N., M.J.G., M.M.F., and J.R.P. Investigation: A.J.S., A.R.M., C.R.G., M.M.F., and J.R.P. Visualization: A.J.S., A.R.M., M.J.G., and J.R.P. Funding acquisition: J.R.P. Project administration: J.R.P. Supervision: J.R.P. Writing—Original draft: A.R.M. and J.R.P. Writing—Review and editing: A.J.S., A.R.M., C.R.G., E.N., M.J.G., M.M.F., and J.R.P.

Competing interests: The authors declare that they have no competing interests.

Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.