Entangling distant resonant exchange qubits via circuit quantum electrodynamics

Donor electron spin qubits hosted within nanoscale devices have demonstrated seconds-long relaxation times at magnetic fields suitable for the operation of spin qubits in silicon of $B=1.5\phantom{\rule{0.16em}{0ex}}\mathrm{T}$. The relaxation rates of these qubits have been shown at milliKelvin temperatures to be mediated by spin-orbit coupling with a ${B}^{5}$ dependency on magnetic field for $B>3\phantom{\rule{0.16em}{0ex}}\mathrm{T}$ with a transition to a ${B}^{3}$ dependency at magnetic fields below ($B\ensuremath{\le}3\phantom{\rule{0.16em}{0ex}}\mathrm{T}$). This deviation has been observed in many spin qubit systems but is particularly notable in multidonor quantum dot qubits. The reason for this has remained a mystery. In this paper we show that for these multidonor low noise, crystalline qubits this deviation at low magnetic fields can be explained by a hyperfine mediated relaxation mechanism of the electron spin through a quantitative model of the relaxation rates. This model identifies the importance of donor nuclear spin flips which are more apparent in multidonor systems in which the larger numbers of donor nuclei creates stronger confinement potentials and enhanced hyperfine couplings. We show theoretically that with atomic precision engineering of the locations of the donor nuclei in these multidonor quantum dot qubits, and/or nuclear spin control, we can minimize the hyperfine mediated relaxation allowing ${T}_{1}$ to extend to $\ensuremath{\sim}200$ seconds ($B=1.5\phantom{\rule{0.16em}{0ex}}\mathrm{T}$).

Recent progress in atomistic modelling and simulations of donor spin qubits in silicon

The recent decade has witnessed substantial advancements in silicon quantum computing. Important milestones include demonstrations of quantum gates exceeding the fault-tolerance threshold, high-fidelity single-shot spin readout, hot quantum bits (hot qubits), and compact scalable spin arrays. Silicon qubits hold promise to leverage semiconductor industry technologies into scalable qubit manufacturing. Both the academic and industry communities are striving to push this advantage into reality. However, formidable challenges persist in the quest to develop a fully operational universal quantum computer. This review focuses on single-spin qubits in silicon. First, we start with foundational spin qubit theory. Then, we discuss gate-defined quantum dots and donor dot systems, with a particular emphasis on two-qubit gate operations and the scalability of qubit arrays. Lastly, we address long-distance coupling, highlighting key areas for future research and potential scale-up strategies for this rapidly evolving field.

We investigate the effect of the valley degree of freedom on Pauli-spin blockade readout of spin qubits in silicon. The valley splitting energy sets the singlet-triplet splitting and thereby constrains the detuning range. The valley phase difference controls the relative strength of the intra- and inter-valley tunnel couplings, which, in the proposed Pauli-spin blockade readout scheme, couple singlets and polarized triplets, respectively. We find that high-fidelity readout is possible for a wide range of phase differences, while taking into account experimentally observed valley splittings and tunnel couplings. We also show that the control of the valley splitting together with the optimization of the readout detuning can compensate the effect of the valley phase difference. To increase the measurement fidelity and extend the relaxation time we propose a latching protocol that requires a triple quantum dot and exploits weak long-range tunnel coupling. These opportunities are promising for scaling spin qubit systems and improving qubit readout fidelity

Phosphorus donor impurities in silicon are a promising candidate for solid-state quantum computing due to their exceptionally long coherence times and high fidelities. However, individual addressability of exchange coupled donors with separations ~15 nm is challenging. We show that by using atomic precision lithography, we can place a single P donor next to a 2P molecule 16 ± 1 nm apart and use their distinctive hyperfine coupling strengths to address qubits at vastly different resonance frequencies. In particular, the single donor yields two hyperfine peaks separated by 97 ± 2.5 MHz, in contrast to the donor molecule that exhibits three peaks separated by 262 ± 10 MHz. Atomistic tight-binding simulations confirm the large hyperfine interaction strength in the 2P molecule with an interdonor separation of ~0.7 nm, consistent with lithographic scanning tunneling microscopy images of the 2P site during device fabrication. We discuss the viability of using donor molecules for built-in addressability of electron spin qubits in silicon.

We address the experimentally relevant problem of robust mitigation of dephasing noise acting on a qubit. We first present an extension of the method of Kuopanportti et al. [Phys. Rev. A 77, 032334 (2008)] for representing $1/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$ noise to the efficient representation of arbitrary Markovian noise. We then add qubit control pulses to enable the design of numerically optimized, two-dimensional, bounded amplitude control functions capable of decoupling the qubit from the dephasing effects of a broad variety of Markovian noise spectral densities during one- and two-qubit quantum operations. We illustrate the method with development of numerically optimized control pulse sequences that minimize decoherence due to a combination of $1/\ensuremath{\omega}$ and constant-offset noise sources. Comparison with the performance of standard dynamical decoupling protocols shows that the numerically optimized pulse sequences are considerably more robust with respect to zero-frequency noise. Application to the mitigation of dephasing noise on spin qubits in silicon indicates that high-fidelity quantum gates may, in principle, be implemented for such qubits with the assistance of current pulse-generation technology.

Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction1. One of the most promising error correction codes is the surface code2, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent3. Among the many qubit platforms, only superconducting circuits4, trapped ions5 and nitrogen-vacancy centres in diamond6 have delivered this requirement. Electron spin qubits in silicon7-15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation16. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.

Shuttling spins with high fidelity is a key requirement to scale up semiconducting quantum computers, enabling qubit entanglement over large distances and favoring the integration of control electronics on-chip. To decouple the spin from the unavoidable charge noise, state-of-the-art spin shuttlers try to minimize the inhomogeneity of the Zeeman field. However, this decoupling is challenging in otherwise promising quantum computing platforms such as hole spin qubits in silicon and germanium, characterized by a large spin-orbit interaction and an electrically tunable qubit frequency. In this work, we show that, surprisingly, the large inhomogeneity of the Zeeman field stabilizes the coherence of a moving spin state, thus also enabling high-fidelity shuttling in these systems. We relate this enhancement in fidelity to the deterministic dynamics of the spin that filters out the dominant low-frequency contributions of the charge noise. By simulating several different scenarios and noise sources, we show that this is a robust phenomenon generally occurring at large field inhomogeneity. By appropriately adjusting the motion of the quantum dot, we also design realistic protocols enabling faster and more coherent spin shuttling. Our findings are generally applicable to a wide range of setups and could pave the way toward large-scale quantum processors. Published by the American Physical Society 2024

Spin qubits in silicon are strong contenders for the realization of a practical quantum computer, having demonstrated single- and two-qubit gates with fidelities above the fault-tolerant threshold, and entanglement of three qubits. However, maintaining high-fidelity operations while increasing the qubit count remains challenging and therefore only two-qubit algorithms have been executed. Here we utilize a four-qubit silicon processor with all control fidelities above the fault-tolerant threshold and demonstrate a three-qubit Grover’s search algorithm with a ~95% probability of finding the marked state. Our processor is made of three phosphorus atoms precision-patterned into isotopically pure silicon, which localise one electron. The long coherence times of the qubits enable single-qubit fidelities above 99.9% for all qubits. Moreover, the efficient single-pulse multi-qubit operations enabled by the electron–nuclear hyperfine interaction facilitate controlled-Z gates between all pairs of nuclear spins with fidelities above 99% when using the electron as an ancilla. These control fidelities, combined with high-fidelity non-demolition readout of all nuclear spins, allow the creation of a three-qubit Greenberger–Horne–Zeilinger state with 96.2% fidelity. Looking ahead, coupling neighbouring nuclear spin registers, as the one shown here, via electron–electron exchange may enable larger, fault-tolerant quantum processors.

Real-time sensing of (spin-dependent) single-electron tunneling is fundamental to electrical readout of qubit states in spin quantum computing. Here, we demonstrate the feasibility of detecting such single-electron tunneling events using an atomically planar charge sensing layout, which can be readily integrated in scalable quantum computing architectures with phosphorus-donor-based spin qubits in silicon (Si:P). Using scanning tunneling microscopy (STM) lithography on a Si(001) surface, we patterned a single-electron transistor (SET), both tunnel and electrostatically coupled to a coplanar ultrasmall quantum dot, the latter consisting of approximately four P donors. Charge transitions of the quantum dot could be detected both in time-averaged and single-shot current response of the SET. Single electron tunneling between the quantum dot and the SET island on a time-scale (τ ∼ ms) two-orders-of-magnitude faster than the spin-lattice relaxation time of a P donor in Si makes this device geometry suitable for projective readout of Si:P spin qubits. Crucial to scalability is the ability to reproducibly achieve sufficient electron tunnel rates and charge sensitivity of the SET. The inherent atomic-scale control of STM lithography bodes extremely well to precisely optimize both of these parameters.

Based on superconducting qubit coupled to a resonator driven by a strong microwave field, we propose a method to implement two quantum logic gates ([Formula: see text]SWAP and NTCP gates) of one qubit simultaneously controlling [Formula: see text] qubits selected from [Formula: see text] qubits in a circuit QED [Formula: see text] by introducing qubit–qubit interaction. The interaction between the qubits and the circuit QED can be achieved by tuning the gate voltage and the external flux. The operation times of the logic gates are much smaller than the decoherence time and dephasing time. Moreover, the numerical simulation under the influence of the gates operations shows that the scheme could be achieved efficiently with presently available techniques.

We theoretically study the transparency and amplification of a weak probe field applied to the cavity in hy- brid systems formed by a driven superconducting circuit QED system and a mechanical resonator, or a driven optomechanical system and a superconducting qubit. We find that both the mechanical resonator and the su- perconducting qubit can result in the transparency to a weak probe field in such hybrid systems when a strong driving field is applied to the cavity. We also find that the weak probe field can be amplified in some parameter regimes. We further study the statistical properties of the output field via the degrees of second-order coherence. We find that the nonclassicality of the output field strongly depends on the system parameters. Our studies show that one can control single-photon transmission in the optomechanical system via a tunable artificial atom or in the circuit QED system via a mechanical resonator.

The evenly spaced modes of an electromagnetic resonator are coupled to each other by appropriate time modulation, leading to dynamics analogous to those of particles hopping between different sites of a lattice. This substitution of a real spatial dimension of a lattice with a ``synthetic'' dimension in frequency space greatly reduces the hardware complexity of an analog quantum simulator. Complex control and readout of a highly multimoded structure can thus be accomplished with very few physical control lines. We demonstrate this concept with microwave photons in a superconducting transmission line resonator by modulating the system parameters at frequencies near the resonator's free spectral range and observing propagation of photon wave packets in time domain. The linear propagation dynamics are equivalent to a tight-binding model, which we probe by measuring scattering parameters between frequency sites. We extract an approximate tight-binding dispersion relation for the synthetic lattice and initialize photon wave packets with well-defined quasimomenta and group velocities. As an example application of this platform in simulating a physical system, we demonstrate Bloch oscillations associated with a particle in a periodic potential and subject to a constant external field. The simulated field strongly affects the photon dynamics despite photons having zero charge. Our observation of photon dynamics along a synthetic frequency dimension generalizes immediately to topological photonics and single-photon power levels, and expands the range of physical systems addressable by quantum simulation.

We present the realization of a cavity quantum electrodynamics setup in which photons of strongly different lifetimes are engineered in different harmonic modes of the same cavity. We achieve this in a superconducting transmission line resonator with superconducting qubits coupled to the different modes. One cavity mode is strongly coupled to a detection line for qubit state readout, while a second long lifetime mode is used for photon storage and coherent quantum operations. We demonstrate sideband-based measurement of photon coherence, generation of n photon Fock states and the scaling of the sideband Rabi frequency with square root of n using a scheme that may be extended to realize sideband-based two-qubit logic gates.

: This is a final report for the IARPA MQCO Spin Qubits team ending in Dec. 2012, when the lead PI, Charles Marcus, moved from Harvard University to University of Copenhagen to direct the Center for Quantum Devices there. The Spin Qubits team comprises experimental and theoretical physicists, and materials scientists, working to realized gate confined multi-qubit systems in GaAs heterostructures. At the completion of the phase covered by this report, two versions of two-qubit systems were in hand, based on both singlet-triplet qubits and Loss-DiVincenzo qubits, but a successful system of three or more qubits had not been realized. The resonant exchange qubit, which allows resonant and single-gate two-qubit operation, and also been demonstrated at the one qubit level, but not for two or more qubits.