Researchers Propose Dense Qubits Using Spin Nonlinearities

Nanomechanical structures are being investigated as a means of achieving long-lived quantum excitations at radio frequencies. Their high quality factors are particularly intriguing as a medium for bosonic encoding of quantum information. However, mechanical modes typically lack the key nonlinearities needed to achieve interaction between bosonic channels, limiting their ability to scale to the many-qubit regime vital for practical quantum computing. Hamza Raniwala of the Massachusetts Institute of Technology, Santa Barbara, and colleagues propose and describe a new approach for bosonic quantum information processing that uses strain-sensitive solids.

High fidelity cat state generation and parity readout using strain-sensitive solid-state spins

Dispersive preparation of even/odd cat states now exceeds a fidelity of 0.98, a substantial improvement over previous methods reliant on larger, less coherent systems such as transmons. This breakthrough surpasses a critical threshold for implementing strong bosonic error correction, previously hindered by the limitations of creating and maintaining these delicate quantum states. Utilising strain-sensitive solid-state spins, the architecture enables the creation of nonclassical mechanical states necessary for advanced quantum computation.

Single-shot optical spin readout of cat-state parity achieves a fidelity of 0.95, providing a reliable method for both state preparation and vital parity checks for error mitigation. This high-fidelity readout is essential for stabilising quantum information and performing complex calculations. These tiny vibrating structures, integrating strain-sensitive solid-state spins with nanomechanical resonators, demonstrate a pathway to high-density quantum computing because they possess exceptionally high quality factors, sustaining quantum behaviour for extended periods.

The architecture enables nearest-neighbour connectivity between mechanical qubits, essential building blocks for larger quantum processors, and outlines the necessary control and readout systems for universal quantum computation. Simulations reveal that interleaving refocusing pulses during parity readout effectively cancels quasi-static noise, maintaining fidelity even with resonator populations of 1.58 and quality factors of 107. Despite these results indicating significant progress towards scalable quantum information processing, they do not yet account for the complexities of fabricating and controlling many of these interconnected components with sufficient precision for fault-tolerant computation.

On-chip architecture for scalable quantum computation using mechanical cat-state qubits

State spins are utilised as nonlinear elements to produce the relevant nonclassical mechanical states. This architecture can allow for a high spatial density of logical qubits by using both the efficiency of bosonic error correction schemes and the small sizes of the constituent nanomechanical resonators and spin qubits. Identifying the necessary performance metrics that will enable error-correction thresholds at high qubit densities clarifies a path towards scalable quantum information processing.

Quantum information processing is predicated on using long-lived quantum modalities in an error-correctable computation scheme to achieve a quantum circuit. To date, large-scale efforts to achieve quantum error correction have focused on implementing surface codes, which have demonstrated sub-threshold error correction for increasing code distance, d. However, surface codes are resource intensive, requiring approximately d2 physical qubits per logical qubit, which causes the quantum information density to be low. Such resource-intensive schemes will likely cause a bottleneck to scaling.

An alternative approach is to perform bosonic encoding, where the infinite-dimensional Hilbert space of a harmonic oscillator is used to encode information. Typical approaches implement these bosonic modes in systems such as microwave electromagnetic cavities, coupled to nonlinear superconducting circuits like transmons to provide the requisite non-classical manipulation. These modalities have recently enabled quantum error correction schemes with logical information lifetimes greater than the physical lifetime of the mode.

However, microwave circuitry is costly and relevant decay timescales; the three-dimensional microwave cavities needed to achieve lifetimes greater than milliseconds are cm3 in dimension, and the nonlinear ancilla transmons have typical dephasing times on the order of tens of microseconds. This limits prospects for scaling beyond the NISQ regime, as the transmon must be coherent during cavity operations, leaving a narrow time window for performing quantum operations. In contrast to the large dimensions required to house a microwave radiation mode, mechanical resonators in the GHz frequency range can be μm3 in dimension.

The acoustic velocity in most materials is roughly four orders of magnitude smaller than the speed of light in a vacuum, meaning a relatively large mechanical mode would occupy less space on-chip than a microwave transmission line resonator with lesser demonstrated lifetimes. Despite this, mechanical modes lack a native nonlinearity to first or second order in most materials, making preparation of a computationally useful nonclassical state difficult. Piezoelectrics and superconducting transmons can circumvent this issue, as has been demonstrated.

However, transmons suffer in both size constraints and qubit lifetimes, limiting the efficacy of a space-efficient platform, and heterogeneous integration of superconductors and piezoelectrics into a low-loss mechanical system is an engineering challenge. Therefore, the prospect of using mechanical resonators coupled to strain-sensitive atomic defect spins in solid state as the ancillary qubits for quantum state preparation is considered.

Unlike transmons, the spin degrees of freedom of atomic defects are inherently long-lived, with population lifetimes on the order of seconds, pure dephasing times surpassing 100μs, and dynamically decoupled dephasing times approaching ms. Consequently, a spin coupled to a mechanical resonator can allow for many more quantum operations on a mechanical mode within its decay time than a transmon would allow for an analogous system.

This concept inverts previous proposals which used spins as the encoding qubit and mechanical modes as a mediating interaction, confining logical quantum information to a much smaller device area. This work develops a unit cell, consisting of a nanomechanical oscillator and ancilla defect spin, capable of the full set of operations required for cat-qubitbased bosonic computation.

Numerical demonstrations, via Lindblad and quantum Monte Carlo simulation of realistic device parameters, show (i) dispersive preparation of even/odd cat states with fidelity exceeding 0.98; (ii) single-shot optical spin readout of cat-state parity with fidelity exceeding 0.95, used both to herald state preparation and to perform stabilising parity checks; (iii) single-qubit Z and X logical gates implemented via dispersive phase accumulation and echoed conditional displacement, respectively; and (iv) spin-mediated twoqubit entangling operations, i.e. heralded Bell-pair generation and a CNOT gate between neighboring mechanical qubits with fidelities up to 0.

Building an explicit error budget for the resulting parity-check cycle, the spin and resonator coherence properties required to exceed a circuit-level error-correction threshold are identified, and it is shown that the long intrinsic coherence of diamond spin ancillae, rather than the mechanical quality factor, is a non-limiting resource in this budget, in contrast to transmon-based bosonic platforms where ancilla dephasing dominates.

Because each logical qubit occupies only the micron-scale footprint of a single nanomechanical resonator, this architecture offers a path to information densities substantially exceeding those of microwave-cavity bosonic codes or superconducting surface codes, confining logical quantum information to a much smaller device area. The platform is evaluated against the DiVincenzo criteria for a viable quantum computing architecture, addressing definition of physical qubit resources, definition of a logical qubit from those resources, qubit initialisation, readout of the logical qubit state, a universal gate set, and coherence times sufficient for fault-tolerant operation.

Each unit cell also includes several classical control elements. Two piezoelectric input ports coupled to rA and rB, respectively, populate the phononic resonators via a small electromechanical interaction driven by a large, tunable driving field. Optical ports allow measurement of the spin ancillae for effective non-destructive readout of the phononic resonators. A superconducting control loop for each ancillary spin crucially allows tuning of the spin transitions.

Quasi-static currents in the superconducting wires allow for tuning of the spin energy by up to ∼300MHz via the Zeeman effect, limited by the critical current density of the superconductor (for Nb, 105 −106A/cm2), while resonant microwaves allow for spin state manipulation. These control elements, piezoelectric driving, optical spin readout, and superconducting control, give us the necessary functionality within a single unit cell. The unit cell Hamiltonian, with energy levels labelled, consists of the bare Hamiltonian Hcell,0 and interaction Hamiltonian Hcell,int.

In the limit where the interaction between rA and rB is small due to low spatial mode overlap, the coupling terms gA,B ∼gB,A ∼The crosstalk between rA and rB is further suppressed by the frequency mismatch between the resonant modes. Critically, the tunability of the spin ancillae allows us to move between the Jaynes-Cummings and dispersive regimes, providing full spin-mediated quantum control of the mechanical resonators as bosonic logical qubits.

Strain-sensitive spins and nanomechanical resonators enable a novel quantum computing architecture

The pursuit of scalable quantum computers hinges on squeezing more qubits into ever-smaller spaces, but achieving this density without sacrificing coherence remains a formidable challenge. Researchers, led by David Awschalom, propose a compelling solution by integrating nanomechanical resonators with strain-sensitive solid-state spins, effectively inverting conventional approaches where spins typically serve as the encoding qubit. This reliance on spin ancillae introduces a new tension: maintaining the long coherence times of these spins while simultaneously achieving strong and reliable coupling to the mechanical resonators.

Acknowledging that maintaining both spin coherence and strong mechanical coupling presents a significant hurdle, this work remains valuable due to its new approach to qubit design. By utilising strain-sensitive spins to create the necessary nonlinearities for mechanical quantum systems, the researchers circumvent limitations hindering scalability. This architecture promises higher qubit density by combining efficient error correction with the small size of the components, representing a key step towards building practical quantum computers.

This research demonstrated a method for bosonic quantum information processing using nanomechanical resonators and strain-sensitive solid-state spins. The integration of these components allows for the creation of nonclassical mechanical states, addressing a key limitation in scaling quantum computing systems. This architecture enables a potentially high density of logical qubits through efficient error correction and the small size of the resonators and spin qubits, operating at resonator frequencies around 2GHz. The authors suggest this work identifies performance metrics needed to reach error-correction thresholds at increased qubit densities, paving a path towards scalable quantum information processing.