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Hardware-efficient quantum random access memory with hybrid quantum acoustic systems
Connor T. Hann, Chang-Ling Zou, Yaxing Zhang, Yiwen Chu, Robert J. Schoelkopf, Steven M. Girvin, Liang Jiang
TL;DR
The paper addresses hardware-efficient QRAM and quantum computing with multimode acoustic systems. It stores quantum information in acoustic modes and uses off-resonant transmon drives to engineer virtual interactions, which can yield higher fidelity than direct gates for long-lived phonons. Parallel operation improves efficiency, but unwanted resonances increasingly constrain scalability as drive counts grow.
Problem
The paper addresses how to build hardware-efficient QRAM and scalable quantum computing architectures with multimode acoustic systems.
Method
Quantum information is stored in acoustic modes while off-resonant drives on an ancillary transmon engineer virtual interactions between modes.
Results
Virtual gates can achieve higher fidelity than direct gates for sufficiently long-lived phonon modes, provided optimal coupling rates are reached.
Takeaways & Limitations
Virtual gates support parallel operation and potentially high connectivity, making the architecture efficient within the proposed multimode setting.
Takeaways & Limitations
Parallelization is limited by the need to check drive-frequency combinations and avoid unwanted resonant couplings as the number of drives grows.
Abstract
from arXiv · showhide
Hybrid quantum systems in which acoustic resonators couple to superconducting qubits are promising quantum information platforms. High quality factors and small mode volumes make acoustic modes ideal quantum memories, while the qubit-phonon coupling enables the initialization and manipulation of quantum states. We present a scheme for quantum computing with multimode quantum acoustic systems, and based on this scheme, propose a hardware-efficient implementation of a quantum random access memory (QRAM). Quantum information is stored in high-Q phonon modes, and couplings between modes are engineered by applying off-resonant drives to a transmon qubit. In comparison to existing proposals that involve directly exciting the qubit, this scheme can offer a substantial improvement in gate fidelity for long-lived acoustic modes. We show how these engineered phonon-phonon couplings can be used to access data in superposition according to the state of designated address modes--implementing a QRAM on a single chip.
Supplemental Material: Hardware-efficient quantum random access memory with
The supplemental material introduces the virtual coupling-rate analysis and verifies its expressions against numerical Floquet calculations.
- The supplemental analysis studies virtual coupling rates and compares their analytical expressions with numerical Floquet results.The comparison is used to assess the accuracy of the derived expressions.
A. Derivation of the virtual coupling rates
The virtual coupling-rate derivation removes qubit–phonon and drive terms perturbatively, yielding resonant phonon interactions under frequency-matching conditions.
- A. Derivation of the virtual coupling rates: Unitary transformations eliminate the qubit–phonon couplings and drive terms in the dispersive, weak-drive regime.The derivation uses λ_k = g_k/δ_k and ξ_j = Ω_j/δ_j, with λ_k ≪ 1.
- A. Derivation of the virtual coupling rates: The rotating-wave approximation neglects rapidly oscillating terms when drive–mode detunings are sufficiently large, while initially omitting phonon Stark shifts.The stated condition is |δj − δk| ≫ λkΩj.
- A. Derivation of the virtual coupling rates: Two drive tones produce a resonant beamsplitter coupling when ω2 − ω1 = ωB − ωA.This condition converts phonons between modes A and B.
- A. Derivation of the virtual coupling rates: A single drive produces a resonant three-mode coupling when its frequency satisfies ω1 = ωA + ωC − ωB.
B. Corrections to the virtual coupling rates
The coupling-rate corrections account for perturbative higher-order terms and nonperturbative AC Stark shifts, which can substantially affect driven virtual interactions.
- B. Corrections to the virtual coupling rates: The Hamiltonian contains corrections β(1,2) beyond the resonant terms, including perturbative contributions and nonperturbative AC Stark-shift effects.The analysis uses the rotating frame of the Stark-shifted qubit to derive the latter contributions.
- B. Corrections to the virtual coupling rates: The Hamiltonian catalog organizes the terms generated by the transformed multimode cQAD Hamiltonian across all drives and modes.The transmon mode is included with λq = 1 and δq = 0.
- B. Corrections to the virtual coupling rates: For the CZ operation, the perturbative correction can significantly reduce the coupling rate when δ1 and δB are comparable to α.For the far-detuned SWAP operation, this correction is typically negligible.
- B. Corrections to the virtual coupling rates: The corrected expressions replace detunings δ with Stark-shifted detunings ˜δ and modify the inverse-Purcell enhancement through β(γ) = (δ/˜δ)^2 − 1.The resulting relation is κγ = κ + γ(g/˜δ)^2.
C. Comparison with numerical Floquet calculation
Analytical virtual coupling rates agree with numerical Floquet calculations for weak drives, while AC Stark shifts explain red-shifted resonances and restrict stronger-drive operation.
- C. Comparison with numerical Floquet calculation: Good agreement between analytical and numerical coupling rates is observed for weak drives, while stronger drives produce expected perturbative discrepancies.For the plotted parameters, agreement holds for ξ ≲ 0.4.
- C. Comparison with numerical Floquet calculation: Including AC Stark corrections red-shifts both analytical and numerical resonances, whereas uncorrected expressions systematically overestimate rates for blue-detuned phonon modes.
- C. Comparison with numerical Floquet calculation: The AC Stark shift produces non-monotonic coupling-rate dependence on drive strength by moving the qubit away from the phonon modes.This behavior restricts useful drive amplitudes during gate optimization.
- C. Comparison with numerical Floquet calculation: Strong drives are not used conservatively because they can induce multiphoton resonances and other deleterious processes.Numerical calculations suggest that coupling could increase beyond the critical drive strength.
- C. Comparison with numerical Floquet calculation: The corrected expressions accurately describe virtual coupling rates for the drive strengths considered in the work.
II. ENGINEERING NONUNIFORM MODE SPACING
Nonuniform mode spacing distinguishes resonance conditions so selected phonon modes can be coupled without degeneracies. The section defines this nonuniformity and its relationship to frequency selectivity and system connectivity.
- II. ENGINEERING NONUNIFORM MODE SPACING: Nonuniform spacing means at least two mode-pair frequency differences νij differ, distinguishing the resonance conditions used for multimode coupling.The frequency spacing is defined as νij = |ωi − ωj|.
- II. ENGINEERING NONUNIFORM MODE SPACING: Frequency selectivity requires the chosen coupling resonance to remain detuned from other resonance conditions by at least ∆ν.Highly selective virtual couplings require gv/∆ν ≪1.
- II. ENGINEERING NONUNIFORM MODE SPACING: Nonuniformity is classified here as point-defect or periodic, based on how successive mode spacings νj,j+1 vary.Point defects vary near one defect, whereas periodic nonuniformities repeat across the spectrum.
A. External mode hybridization
External-mode hybridization creates local frequency nonuniformity that enables selective coupling of phonon modes. Its usefulness is limited by the finite affected bandwidth and the external mode’s coherence.
- A. External mode hybridization: Coupling phonons to an external mode shifts frequencies within bandwidth D, creating nonuniformity that can make selected two- and three-mode resonances nondegenerate.The resulting frequency shifts determine ∆ν; shifts of order 1 MHz have been demonstrated.
- A. External mode hybridization: Selective two-mode coupling is possible when at least one involved mode lies within D, while selective three-mode coupling requires two involved modes there.Modes outside D cannot be directly coupled to one another under this strategy.
- A. External mode hybridization: The external mode should have phonon-comparable coherence because hybridization can otherwise increase effective decay and reduce gate fidelity.Greater nonuniformity may trade off against enhanced decay.
B. Two phonon mode families
Using two phonon mode families with different free spectral ranges creates periodic nonuniformity that supports selective cross-family coupling. Three-mode selectivity requires an additional external mode or a restricted coupling bandwidth, and example SAW and BAW implementations are described.
- B. Two phonon mode families: Different free spectral ranges make the spacing between modes from two families vary, enabling selective coupling between modes from different families.Selectivity is guaranteed only within a finite region S.
- B. Two phonon mode families: Two mode families alone do not enable selective three-mode coupling, but an external mode can provide the missing coupling pathway.A high-Q cavity, another resonator, or a restricted transmon-phonon bandwidth can supply the additional mode or selectivity.
- B. Two phonon mode families: Composite resonators offer another periodic-nonuniformity strategy through interface reflections that modulate the free spectral range.The resulting selectivity and bandwidth depend on the modulation’s nature.
- B. Two phonon mode families: Example devices engineer nonuniformity using external-mode hybridization in SAW systems or two BAW mode families coupled through a 3D transmon.The SAW design includes a coplanar waveguide resonator, while the BAW design uses additional piezoelectric transduction.
B. Database access schemes
The QRAM supports distinct access procedures for quantum, classical, and classical read-only databases. Quantum data is accessed through pointer-controlled swaps, whereas classical data can be copied nondestructively to a bus using phase operations.
- B. Database access schemes: Quantum data cannot be copied during a query, so the bus must instead be entangled or swapped with the selected database data.Classical data does not have this no-cloning constraint.
- B. Database access schemes: Quantum database access routes a pointer qubit along the address-selected path and extracts the selected phononic data through controlled-SWAP operations.The data, pointer, and address qubits are then routed out sequentially.
- B. Database access schemes: Classical data encoded in phonon Fock states can be copied to a bus qubit with a CZ operation without disturbing other modes.The bus starts in |+⟩, and the selected bit is represented in the |±⟩ basis.
- B. Database access schemes: Classical read-only data can remain in a purely classical memory array and be transferred to the bus through entry-dependent phase shifts.This avoids storing the read-only database in phonon-mode Fock states.
IV. DERIVATION OF GATE FIDELITY EXPRESSIONS
The section defines local and global gate fidelities and derives infidelity contributions from decoherence and spectral crowding. It combines these contributions into approximations whose plotted forms include higher-order corrections.
- Global fidelity explicitly accounts for crosstalk across all M stored qubits, whereas local fidelity evaluates effects on only the qubits involved in the gate.The global metric generally depends on M, while the local metric is largely independent of implementation differences.
- Decoherence and spectral crowding are the two dominant contributions to both direct and virtual gate infidelities.The decoherence terms depend on gate times and the relevant direct or virtual decoherence rates.
- Virtual-gate spectral crowding is determined by unwanted couplings involving at least one mode participating in the gate, with detunings set by the nonuniform mode structure.For two-mode gates, the relevant detunings are differences between the desired transition and nearby unwanted transitions.
- The spectral-crowding contribution scales as (g_v/∆ν)^2 regardless of the specific nonuniformity structure.The detuning pattern affects numerical coefficients, but not this scaling.
- The first-order infidelity expressions apply only when 1 − F ≪ 1, so plots use higher-order expressions constrained to the interval [0,1].The plotted form is 1 − e^−kt(1 − ε), rather than the first-order approximation kt + ε.
B. Global infidelity
Global infidelity includes decoherence and spectral crowding across all stored modes, making it dependent on the total number of modes and their frequency distribution. For the studied BAW/SAW implementation, increasing M reduces both direct and virtual fidelities, with stronger reductions for virtual gates.
- Global spectral-crowding infidelity requires the full transition-frequency distribution and is evaluated here only for a particular BAW/SAW implementation.The considered device uses approximately uniform direct-gate mode spacing and two mode families with different free spectral ranges for virtual gates.
- The direct spectral-crowding coefficient is S_d = π^2/3 and is independent of whether other modes store qubits.Unwanted transmon–phonon transitions contribute to direct-gate infidelity regardless of the encoding of other modes.
- The virtual spectral-crowding coefficient S_v increases with M at fixed ν/∆ν.This coefficient averages unwanted virtual couplings affecting at least one of the M stored qubits.
- For M = 10, both direct and virtual global gate fidelities are reduced relative to M = 2, with the virtual reduction more pronounced.The stronger virtual reduction is partly attributed to the growth of the virtual spectral-crowding coefficient with M.
- The global comparison is platform-dependent because spectral-crowding coefficients can scale differently with M in phononic-crystal devices.In the discussed phononic-crystal case, the effective free spectral range scales as 1/M and S_d scales as M^2.
- The analysis recommends modular devices for large M because controlling all modes with one transmon becomes impractical, with roughly 10 modes per module favored over 100 or 1000.For the BAW/SAW device, the quantum-volume optimum satisfies M ≤ 12 across the plotted parameter space.
1. Effects of Kerr on two-mode couplings
Self- and cross-Kerr interactions perturb engineered two-mode couplings through state-dependent detunings. Under experimentally motivated coupling and Kerr strengths, their infidelity is typically much smaller than decoherence and spectral-crowding contributions.
- Two-mode Kerr interactions produce state-dependent shifts that cannot generally be compensated because phonon occupations are not known a priori.The resulting detuning depends on cross-Kerr differences between the participating and other modes.
- For demonstrated parameters g_v/2π = 30 kHz and χ/2π ≲ 1 kHz, the Kerr-related infidelity contribution is < 10^−4.These values come from an experimentally demonstrated beamsplitter-type coupling between two microwave cavity modes mediated by a transmon.
- For parameters used in the main-text analysis, typical cross-Kerr differences of about 100 Hz to 1 kHz yield infidelities of 10^−5 − 10^−3.The estimate assumes a total phonon number usually of order 10 or fewer.
2. Effects of Kerr on three-mode couplings
Three-mode Kerr interactions modify the resonance condition for engineered couplings, but drive-frequency compensation can remove the shifts from the participating modes. Residual infidelity depends on deviations in cross-Kerr couplings and can be further reduced with mode-dependent compensation.
- Without compensation, cross-Kerr interactions with other modes can produce detuning based on the couplings themselves rather than their differences, potentially making the infidelity significant.The relevant contribution scales as (g_v^(2)/D^(2))^2.
- Residual three-mode detuning depends only on deviations from the average cross-Kerr coupling, yielding an estimated infidelity of 10^−4 − 10^−2.Mode-dependent compensation can suppress this contribution further.
- Drive-frequency compensation removes AC Stark shifts and self- and cross-Kerr terms among modes A, B, and C from the three-mode resonance condition.The compensated condition includes the average cross-Kerr shift and total phonon number.
- The engineered two- and three-mode interactions support universal computation in a dual-rail encoding through arbitrary single-qubit rotations and a CZ gate.The three-mode interaction applies a −1 phase to the logical |1̄1̄⟩ state while leaving the other logical states unaffected.
VII. PARALLELISM AND LIMITATIONS
Virtual gates can execute selected phonon-mode operations in parallel, but parallelization requires frequency-collision checks and is constrained by drive-induced transmon heating.
- Parallel gate example: Parallel virtual gates can implement separate SWAP operations simultaneously when suitable drive tones bring only the intended mode couplings on resonance.A two-SWAP example requires at least three distinct drive tones when the relevant mode-frequency differences differ.
- Parallelization constraints: For d simultaneous drives, d(d −1)/2 pairwise frequency combinations must be checked to prevent unintended resonant couplings.Each drive frequency also requires additional checks against mode-frequency differences used for storing quantum information.
- Heating limitation: Drive-induced transmon heating from dissipation and dephasing can limit operation fidelity as the number of applied drives increases.The heating rate is introduced for two drive tones and depends on drive-related parameters and transmon loss.
- Heating limitation: For the weak drives considered, additional heating contributes negligibly to infidelity relative to the inverse Purcell effect, but stronger parallel driving can increase it.This establishes a practical trade-off between parallel operation and fidelity.