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Entangling Bosonic Modes via an Engineered Exchange Interaction

Yvonne Y. Gao, Brian J. Lester, Kevin Chou, Luigi Frunzio, Michel H. Devoret, Liang Jiang, S. M. Girvin, Robert J. Schoelkopf

arXiv:1806.07401v1quant-ph

TL;DR

The controlled-SWAP operation is limited by the transmon's need to remain coherent during a slow SWAP. The paper simplifies the eSWAP circuit and reports its successful implementation on multi-photon, error-correctable bosonic codewords, while identifying fidelity limitations and possible routes to suppress them.

  • Problem

    A controlled-SWAP requires the transmon to remain in a coherent superposition during the slow SWAP, making its performance vulnerable to transmon T1 and T2 errors.

    Source: THE SIMPLIFIED EXPONENTIAL-SWAP · p. 9

  • Method

    The eSWAP circuit decomposes the controlled-SWAP into two beamsplitters and a controlled phase gate, then uses commutation and cancellation to simplify the sequence.

    Source: THE SIMPLIFIED EXPONENTIAL-SWAP · p. 10

  • Results

    The results show successful eSWAP implementation on two bosonic modes encoded in multi-photon, error-correctable codewords.

    Source: eSWAP ERROR BUDGET · p. 13

  • Takeaways & Limitations

    The gate-fidelity limitations are understood and can be suppressed through improved coherence times and more sophisticated calibration.

    Source: eSWAP ERROR BUDGET · p. 13

    Stronger drives can introduce additional cavity decoherence and spurious transitions, while cavity self-Kerr detunes the bilinear coupling and reduces beamsplitter quality.

    Source: eSWAP ERROR BUDGET · pp. 12–13

Abstract

from arXiv · show

The realization of robust universal quantum computation with any platform ultimately requires both the coherent storage of quantum information and (at least) one entangling operation between individual elements. The use of continuous-variable bosonic modes as the quantum element is a promising route to preserve the coherence of quantum information against naturally-occurring errors. However, operations between bosonic modes can be challenging. In analogy to the exchange interaction between discrete-variable spin systems, the exponential-SWAP unitary [$\mathbf{U}_{\mathrm{E}}\left(\theta_c\right)$] can coherently transfer the states between two bosonic modes, regardless of the chosen encoding, realizing a deterministic entangling operation for certain $\theta_c$. Here, we develop an efficient circuit to implement $\mathbf{U}_{\mathrm{E}}\left(\theta_c\right)$ and realize the operation in a three-dimensional circuit QED architecture. We demonstrate high-quality deterministic entanglement between two cavity modes with several different encodings. Our results provide a crucial primitive necessary for universal quantum computation using bosonic modes.

Device architecture and system parameters

Alice and Bob are 3D superconducting cavity memories in a shared package with three transmons. The central qC provides mixing, while qA and qB support individual cavity control and tomography, with qB also controlling the eSWAP angle.

Device architecture

  • Alice and Bob are coaxial 3D cavity memories with lengths of 4.8 and 5.6 mm, respectively, housed with three transmons and readout resonators in one aluminum block.The cavities are shorted at one end and open at the other through a narrow circular waveguide.Source: Device architecture and system… · p. 7

System layout

  • The top-view schematic places qC between Alice and Bob, with qA and qB coupled to their respective cavities and RF drives entering through qC's drive port.Source: Figure S1 · p. 7

Resonator design

Ancilla roles

  • The Y-shaped qC mixes Alice and Bob, while qA and qB provide independent cavity manipulation and tomography; qB also controls the eSWAP angle.Each of qA and qB is coupled to one cavity, and qC couples to both.Source: Device architecture and system… · p. 7

Joint Wigner tomography and state reconstruction

The section introduces joint Wigner tomography using independent cavity ancillas and reports state-fidelity benchmarks for Fock and binomial encodings. The reported measurements also identify preparation, measurement, and coherence constraints on those benchmarks.

Joint Wigner tomography and state reconstruction

  • The two-ancilla measurement avoids the earlier qC-based approach, which required multiple qC levels and a stringent matched-χ condition for the two cavity modes.Source: Joint Wigner tomography and state… · p. 7
  • Joint single-shot readout of qA and qB gives the product of Alice's and Bob's displaced parities, yielding the joint Wigner function and a density matrix up to the chosen photon cutoff.The cutoff depends on the encoding.Source: Joint Wigner tomography and state… · p. 7

State-fidelity results

  • ∼85% average state fidelity across the 16 {0, 1} Fock basis states with the operation, versus ∼88% without it.For |0⟩A|0⟩B, tomography yields ∼90% state fidelity, consistent with ∼0.95 parity contrast for each cavity; readout errors, ancilla decoherence, and imperfect transmon rotations contribute to the limitation.Source: Joint Wigner tomography and state… · p. 8

State-fidelity analysis

State-fidelity results

  • ∼80% average fidelity for binomial-encoding initial states without eSWAP is attributed primarily to longer optimal-control preparation pulses that can fail if the transmon dephases.The encoding has average photon number n̄ = 2 in each mode, and the cited pulse timescale is n̄/χ ∼ 1 µs.Source: Joint Wigner tomography and state… · p. 8

Limitations

REALIZATION OF A QUANTUM CONTROL-SWAP (FREDKIN) GATE

The experiment uses a transmon-state-dependent resonance to implement a deterministic controlled-SWAP between two cavity modes. Conditional tomography finds the reconstructed states qualitatively agree with ideal states and shows three-mode entanglement for superposition inputs.

REALIZATION OF A QUANTUM CONTROL-SWAP (FREDKIN) GATE

  • Drive-frequency spectroscopy identifies a state-selective SWAP resonance, allowing qB to control whether Alice and Bob exchange states.The SWAP occurs at two separated drive frequencies; the chosen frequency makes the coupling resonant when qB is in |e⟩.Source: REALIZATION OF A QUANTUM CONTROL-SWAP… · pp. 8–9, Figure S2 · p. 9
  • The deterministic Fredkin gate is characterized by conditional tomography of four inputs, reconstructing the joint state of Alice, Bob, and qB.The protocol measures qB in an X, Y, or Z basis, then uses a joint Wigner measurement of the cavities.Source: REALIZATION OF A QUANTUM CONTROL-SWAP… · p. 9
  • For density matrices extracted for four initial states—|g⟩|0⟩|1⟩, |e⟩|0⟩|1⟩, |+⟩|0⟩|1⟩, and |−⟩|0⟩|1⟩—the average overlap with ρideal was 68±5%, without correcting for SPAM errors.The bottom two plots show evidence of entanglement between the three modes, with phases consistent with the initial superposition of qB.Source: REALIZATION OF A QUANTUM CONTROL-SWAP… · p. 9
  • The authors report the first experimental realization of a deterministic quantum Fredkin gate in cQED, a tool for quantum computation, cryptography, and measurement.Source: REALIZATION OF A QUANTUM CONTROL-SWAP… · p. 9

THE SIMPLIFIED EXPONENTIAL-SWAP

The simplified eSWAP circuit reduces gate time and, especially, the time the transmon spends outside its ground state. It achieves this by decomposing and reordering the controlled-SWAP operations so that beamsplitters can be eliminated and the transmon stays in its ground state during them.

THE SIMPLIFIED EXPONENTIAL-SWAP

  • The original eSWAP protocol uses two cSWAP gates and an intervening x-axis transmon rotation, requiring the transmon to remain coherent throughout the slow SWAP operations.The resulting exposure makes performance vulnerable to transmon T1 and T2 errors; stronger drives can add imperfections or unwanted transitions.Source: THE SIMPLIFIED EXPONENTIAL-SWAP · pp. 9–10
  • The simplified circuit is designed to reduce overall gate time and, especially, the transmon's duration outside its ground state.The authors identify the shorter non-ground interval as a way to suppress decoherence effects on eSWAP quality.Source: THE SIMPLIFIED EXPONENTIAL-SWAP · p. 10
  • Decomposing cSWAP into two beamsplitters and a controlled phase gate, then commuting operations, permits canceling sequential beamsplitters and preparing the transmon in its ground state for them.The final circuit places the transmon in a coherent superposition only during the faster controlled phase gate and rotation.Source: THE SIMPLIFIED EXPONENTIAL-SWAP · p. 10, Figure S3 · p. 10

EXTENDED DATA: THE CHI-MATRIX

The χ-matrix analysis gives process fidelities for the identity and eSWAP operations consistent with the Pauli-transfer analysis. It also explains why the paper uses the Pauli-transfer representation for its main analysis.

EXTENDED DATA: THE CHI-MATRIX

  • Process fidelities in the χ representation are 0.88, 0.82, and 0.82 for θ = 0, π/4, and π/2, respectively.The identity process used for state preparation has fidelity ∼0.85.Source: EXTENDED DATA: THE CHI-MATRIX · p. 10
  • Both χ and Pauli-transfer matrices completely represent a quantum process, but the paper uses the real-valued Pauli-transfer matrix R for convenience.χ is complex and over-complete; differing error propagation and statistical uncertainty may explain fidelity differences between representations.Source: EXTENDED DATA: THE CHI-MATRIX · pp. 10–11
  • The χ matrices show real and imaginary components, with measured matrices above ideal matrices for the input, identity, and SWAP operations.The measured matrices are uncorrected for SPAM errors.Source: Figure S4 · p. 11

eSWAP ERROR BUDGET

The error analysis attributes most of the Fock-encoding process infidelity to SPAM and identifies operation errors and photon-number-dependent effects that limit binomial-encoding fidelity. It outlines drive and hardware changes that may improve operation quality, while noting trade-offs.

eSWAP ERROR BUDGET

  • Initial states in Alice and Bob are prepared with numerically optimized GRAPE pulses requiring simultaneous drives on each cavity and transmon for a minimum duration tprep ∼ n̄/χ; during preparation, the system is vulnerable to transmon-ancilla relaxation and dephasing.Source: eSWAP ERROR BUDGET · p. 11
  • Binomial encoding has greater SPAM error than Fock encoding, consistent with lower state-preparation and readout fidelity at larger photon number.The identity-operation process fidelities measured after encoding are ∼88% for Fock and ∼77% for binomial encoding.Source: eSWAP ERROR BUDGET · pp. 11–12
  • The eSWAP operation adds ∼4% infidelity in the {0, 1} Fock encoding, with process fidelity limited primarily by SPAM errors.The authors attribute likely operation errors mainly to qC heating during beamsplitter operations and cavity photon loss, expected to contribute ∼1% infidelity.Source: eSWAP ERROR BUDGET · p. 12
  • At larger photon number, self-Kerr, cavity decoherence, and calibration sensitivity add operation non-idealities; self-Kerr detunes the bilinear coupling and reduces beamsplitter quality.The effect was observed for coherent-state encoding with n̄ ≈2 in each cavity, and the simulation including self-Kerr reproduces the measured Wigner-function features.Source: eSWAP ERROR BUDGET · p. 12, Figure S5 · p. 12
  • Binomial-encoding process fidelities are R0 ≈0.7, Rπ/4 ≈0.58, and Rπ/2 ≈0.65, with encoding fidelity Rencode ≈0.77.The measured matrices show qualitative agreement with ideal processes but reduced contrast; the process results are without SPAM correction.Source: eSWAP ERROR BUDGET · p. 12, Figure S6 · p. 13
  • Strong drives used for tunable beamsplitters can increase photon loss and cavity self-Kerr; without QEC, photon loss reduces fidelity, while the enhanced self-Kerr is not currently QEC-mitigable.Source: eSWAP ERROR BUDGET · pp. 12–13
  • The binomial-encoding error budget estimates ∼5% photon-loss infidelity at two photons per cavity, ∼8% self-Kerr infidelity (5–15%), and ≤3% conditional-phase miscalibration.The table also estimates qC pump-induced heating below ∼2% and ancilla-excitation error below 1%; its ∼1% photon-loss estimate applies to the {0, 1} Fock encoding.Source: eSWAP ERROR BUDGET · p. 13
  • Faster beamsplitters may reduce exposure to photon loss and self-Kerr, but stronger drives can add cavity decoherence and spurious transitions; more sophisticated mixing elements are also proposed.The authors report successful eSWAP on multi-photon, error-correctable codewords and identify improved coherence and calibration as ways to suppress fidelity limitations.Source: eSWAP ERROR BUDGET · p. 13
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