## Building an essential quantum component

To build a universal quantum computer—the kind that can handle any computational task you throw at it—an essential early step is to demonstrate the so-called CNOT gate, which acts on two qubits. Zajac *et al.* built an efficient CNOT gate by using electron spin qubits in silicon quantum dots, an implementation that is especially appealing because of its compatibility with existing semiconductor-based electronics (see the Perspective by Schreiber and Bluhm). To showcase the potential, the authors used the gate to create an entangled quantum state called the Bell state.

## Abstract

Single-qubit rotations and two-qubit CNOT operations are crucial ingredients for universal quantum computing. Although high-fidelity single-qubit operations have been achieved using the electron spin degree of freedom, realizing a robust CNOT gate has been challenging because of rapid nuclear spin dephasing and charge noise. We demonstrate an efficient resonantly driven CNOT gate for electron spins in silicon. Our platform achieves single-qubit rotations with fidelities greater than 99%, as verified by randomized benchmarking. Gate control of the exchange coupling allows a quantum CNOT gate to be implemented with resonant driving in ~200 nanoseconds. We used the CNOT gate to generate a Bell state with 78% fidelity (corrected for errors in state preparation and measurement). Our quantum dot device architecture enables multi-qubit algorithms in silicon.

Gate-defined semiconductor quantum dots are a powerful platform for isolating and coherently controlling single electron spins (*1*, *2*). Silicon quantum dots can take advantage of state-of-the-art industrial nanofabrication capabilities for scalability (*3*), and support some of the longest quantum coherence times measured in the solid state (*4*, *5*). By engineering local magnetic field gradients, electron spins can be electrically controlled (*6*, *7*) with single-qubit gate fidelities exceeding 99.9% (*8*, *9*). Although exchange control of spins was demonstrated as early as 2005 (*10*), demonstrations of two-qubit gates with quantum dot spins are scarce owing to technological and materials challenges (*11*–*14*). A demonstration of an efficient CNOT gate for spins in silicon will open a path for multi-qubit algorithms in a scalable semiconductor system.

Here, we demonstrate a ~200-ns CNOT gate in a silicon semiconductor double quantum dot (DQD), nearly an order of magnitude faster than the previously demonstrated composite CNOT gate (*14*). Turning on an exchange interaction results in a state-selective electron spin resonance (ESR) transition that is used to implement a CNOT gate with a single microwave (MW) pulse. Unlike previous DQD implementations of the exchange gate, our CNOT gate is implemented at a symmetric operating point, where the exchange coupling *J* is first-order insensitive to charge noise (*15*–*17*).

We use the spin of a single electron to encode a qubit (*18*). A gate-defined DQD (Fig. 1A) isolates two electrons in a silicon quantum well with natural isotopic abundance, forming a two-qubit device (Fig. 1B). Gates L and R are used to control the energy of the electrons trapped in the left and right quantum dots, respectively, and gate M provides control of *J*. The charge occupancy of the DQD is detected by monitoring the current *I*_{s} or conductance *g*_{s} through a nearby quantum dot charge sensor (*19*). A Co micromagnet (*20*) generates a magnetic field gradient that results in distinct ESR transition frequencies for the left and right qubits. Electrically driven single-qubit rotations are implemented by “shaking” the electron spins in the transverse field gradient of the micromagnet (*7*, *21*). Other Si/SiGe device designs have suffered from accidental quantum dot formation, which is in part caused by the high effective mass of electrons in Si and by gate designs that are much larger than the single-electron wave function (*22*). In our experiments, isolation and control of single electrons are made possible by an overlapping aluminum gate electrode architecture that provides tight electronic confinement in a largely disorder-free potential. Our devices allow exquisite control of single electrons and nearest-neighbor exchange coupling, with the potential to scale to at least nine dots in a linear array (*19*).

The DQD gate voltages *V*_{L} and *V*_{R} are tuned to traverse the charge stability diagram. Starting from the charge state (*N*_{L} = 0, *N*_{R} = 0), where *N*_{L} and *N*_{R} respectively refer to the number of electrons in the left and right dots, we navigate from point a to point c in the stability diagram (Fig. 1C) to initialize the device in the |↓_{L}↓_{R}〉 state. The gates are then pulsed to point d, in the (1, 1) charge state, where single-qubit control is achieved by applying MW pulses to gate S. Exchange can be rapidly turned on and off (on a 5-ns time scale) by adjusting the voltage *V*_{M}. Qubit readout is achieved by moving from point e to point g, where the left and right dot spins are sequentially measured and emptied. Spin-dependent tunneling and charge state readout are used to extract the spin-up probabilities and of the left and right qubits, respectively, following references (*23*, *24*). Energy level diagrams corresponding to each point in the pulse sequence are shown in Fig. 1D. High-fidelity single-qubit control is demonstrated in Fig. 2, where randomized benchmarking yields single-qubit fidelities *F*_{L} = 99.3 ± 0.2% and *F*_{R} = 99.7 ± 0.1%.

Proposals for two-qubit interactions with spins in semiconductors are generally based on control of the exchange coupling (*18*). To implement a high-fidelity CNOT gate, we must first measure *J* as a function of *V*_{M} (*20*). Physically, in the presence of a strong magnetic field gradient δ*B* >> *J*, the exchange interaction lowers the energy of the antiparallel spin states relative to the |↑↑〉 and |↓↓〉 spin states (Fig. 3A). As a result, the ESR frequency of the left qubit will be dependent on the state of the right qubit, and vice versa. We can therefore determine *J* by measuring the left-qubit ESR spectra for different right-qubit states (Fig. 3B). Specifically, the system is prepared in |↓↓〉 and then a rotation of duration τ_{R} is applied to the right qubit. Next, we apply a low-power probe tone for a time τ_{L} >> *T*_{2} at a frequency *f*_{p} that will leave the qubit in a mixed state if *f*_{p} is resonant with the qubit frequency. For the simple case where τ_{R} is such that the right qubit ends in the spin-down state, the left qubit will have a transition frequency ; likewise, when the right qubit ends in the spin-up state, the left qubit’s transition frequency will be (Fig. 3B, green and blue boxes, respectively). By plotting as a function of τ_{R} and *f*_{p} (Fig. 3C), we see that the left-qubit resonance frequency is correlated with the state of the right qubit (Fig. 3D). The exchange frequency *J*/*h* = , where *h* is Planck’s constant, is directly extracted from the data sets in Fig. 3E and plotted as a function of *V*_{M} in Fig. 3F (*20*). A 20-mV change in *V*_{M} is sufficient to turn on an exchange interaction of 10 MHz, which exceeds typical single-qubit Rabi frequencies (*f*_{Rabi} = 4.8 MHz in Fig. 2C).

Fast gate voltage control of *J* can be used to implement a resonant CNOT gate (Fig. 4). The general quantum circuit and its experimental implementation are shown in Fig. 4, A and B. When *V*_{M} is low, *J* is approximately zero (*J* ~ 300 kHz for *V*_{M} = 390 mV; see Fig. 3F), corresponding to the level diagram at the left in Fig. 4C (*20*). With *J* ~ 0, high-fidelity single-qubit gates can be implemented, because the resonance frequency of each qubit is independent of the state of the other qubit (see Fig. 2). When *V*_{M} is pulsed high, the antiparallel spin states are lowered in energy by *J*/2 relative to the parallel spin states (Fig. 4C, right). In this configuration, the CNOT gate is resonantly implemented by driving the |↓↑〉 to |↑↑〉 transition. Here, |tc〉 describes a product state of the target (t) and control (c) qubits. In contrast, the |↑↓〉 to |↓↓〉 transition is off-resonant as a result of the *J* coupling, allowing for a conditional rotation that is dependent on the right electron’s spin state.

To calibrate the CNOT gate, we use a long direct-current exchange pulse of length τ_{dc} = 1 μs and vary the length τ_{P} of the MW pulse to drive transitions between |↓↑〉 and |↑↑〉. The resulting conditional oscillations are shown in Fig. 4D for the input states |↓↑〉 and |↓↓〉. A conditional π-rotation is realized on the target qubit for *t*_{CNOT} = τ_{P} = 130 ns. Thanks to the magnetic field gradient, changes in *V*_{M} shift the orbital positions of the electrons and result in small changes in the ESR resonance frequencies. By setting τ_{dc} = 2π/*J* = 204 ns, we eliminate conditional phases caused by exchange and the remaining single-qubit phases are accounted for in the phase of the consecutive MW drives, resulting in a pure CNOT gate (*20*). In contrast to our single-step CNOT gate, implementation of a conventional CNOT gate following the Loss-DiVincenzo proposal would require mastery of two operations and three single-qubit gates (*18*).

In general, the CNOT gate must be able to operate on an arbitrary input state, and specifically on product states of the form(1)where R and L denote the right (control) and left (target) qubits. Here, |α_{L,R}|^{2} + |β_{L,R}|^{2} = 1. To test the basic functionality of the CNOT gate, we first initialize the system in |↓↓〉. The control qubit is then rotated by an angle θ_{R} to create the input state(2)Figure 4E shows and measured after the CNOT gate acts on different input states with angle θ_{R}. These data show that the CNOT acts as expected on states outside of the classical set of (product) input states |↑↑〉, |↑↓〉, |↓↑〉, and |↓↓〉 and that the control qubit is not flipped during the CNOT operation.

We next use the CNOT gate (*20*) to create the Bell state |ψ_{target}〉 = and extract its fidelity by performing two-qubit state tomography (*25*, *26*). The Bell state is created by initializing the system in |↓↓〉 and applying a π/2* _{x}* pulse on the control spin, which generates the input state |ψ

_{in}〉 = . Application of the CNOT gate to |ψ

_{in}〉 yields |ψ

_{target}〉 = . State tomography is performed by appending single-qubit rotations after the CNOT gate to measure the expectation value for all two-qubit Pauli operators (for example, by applying a π/2

*rotation to the left qubit and π/2*

_{x}*rotation to the right qubit, we measure the YX two-qubit operator). Because the set of Pauli operators forms a basis of the Hermitian operators on the two-qubit Hilbert space, we can reconstruct the full two-qubit density matrix from these measurements. Not accounting for imperfections in state preparation and measurement (SPAM), we obtain a fidelity*

_{y}*F*

_{raw}= 〈ψ

_{target}|ρ|ψ

_{target}〉 = 56% (

*27*). The readout visibilities of both qubits (

*20*) and spin relaxation during the sequential qubit readout together account for a considerable amount of the infidelity. By adopting the procedure used in (

*22*), we correct our density matrix to account for SPAM errors and obtain a corrected fidelity

*F*= 〈ψ

_{target}|ρ|ψ

_{target}〉 = 78%. We anticipate that the state fidelity can be further improved through the use of optimized pulse sequences (

*28*).

Realizing robust two-qubit gates has been a bottleneck in the development of spin-based quantum computers (*12*, *29*). By combining our results with recent advances in Si/SiGe quantum dot device technology (*19*), we anticipate that it will become feasible to demonstrate nine-qubit quantum processors in silicon.

## Supplementary Materials

www.sciencemag.org/content/359/6374/439/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S10

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank T. Hazard, J. Stehlik, and K. Wang for technical assistance. Research was sponsored by Army Research Office grant W911NF-15-1-0149, the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4535, and NSF grant DMR-1409556. Devices were fabricated in the Princeton University Quantum Device Nanofabrication Laboratory. Any mention of commercial products is for information only; it does not imply recommendation or endorsement by NIST. The data that support the findings of this study are available in the supplementary materials. Additional data can be obtained from the corresponding author upon request.