PaperlayerView exact revision on arXiv

Source-linked AI summary

Deterministic teleportation of a quantum gate between two logical qubits

K. S. Chou, J. Z. Blumoff, C. S. Wang, P. C. Reinhold, C. J. Axline, Y. Y. Gao, L. Frunzio, M. H. Devoret, Liang Jiang, R. J. Schoelkopf

arXiv:1801.05283v1quant-ph

TL;DR

The experiment uses high-fidelity measurement and real-time control to condition operations in a teleported CNOT implementation. It characterizes measurement-basis effects and reports encoding-dependent gate errors and dominant infidelity sources.

  • Problem

    Real-time measurement and control are needed to apply classically conditional operations, while the communication-qubit measurement basis affects the teleported gate's logical phase.

  • Method

    The experiment combines fast transmon measurement, real-time state estimation, conditional π pulses, feedback preparation, and measurement-basis calibration for teleported-gate operation.

  • Results

    Conditioned teleported-gate errors were (16 ± 3)% for Binomial encoding and (13 ± 2)% for Fock encoding, with finite T2 and T1 contributing roughly 70% and 25% of total infidelity.

  • Takeaways & Limitations

    The measurement basis must be calibrated because it induces a logical phase, while transmon coherence dominates the reported teleported-CNOT infidelity.

Abstract

from arXiv · show

A quantum computer has the potential to effciently solve problems that are intractable for classical computers. Constructing a large-scale quantum processor, however, is challenging due to errors and noise inherent in real-world quantum systems. One approach to this challenge is to utilize modularity--a pervasive strategy found throughout nature and engineering--to build complex systems robustly. Such an approach manages complexity and uncertainty by assembling small, specialized components into a larger architecture. These considerations motivate the development of a quantum modular architecture, where separate quantum systems are combined via communication channels into a quantum network. In this architecture, an essential tool for universal quantum computation is the teleportation of an entangling quantum gate, a technique originally proposed in 1999 which, until now, has not been realized deterministically. Here, we experimentally demonstrate a teleported controlled-NOT (CNOT) operation made deterministic by utilizing real-time adaptive control. Additionally, we take a crucial step towards implementing robust, error-correctable modules by enacting the gate between logical qubits, encoding quantum information redundantly in the states of superconducting cavities. Such teleported operations have significant implications for fault-tolerant quantum computation, and when realized within a network can have broad applications in quantum communication, metrology, and simulations. Our results illustrate a compelling approach for implementing multi-qubit operations on logical qubits within an error-protected quantum modular architecture.

CONTRIBUTIONS

The experiment and manuscript were carried out by designated authors spanning experimentation, analysis, control software, fabrication, design, theory, and writing.

  • K.S.C, J.Z.B, and C.S.W. performed the experiment and analyzed the data under R.J.S.’s supervision.
  • P.C.R. developed feedforward control software and implemented software for generating optimal control pulses.
  • C.J.A., Y.Y.G., and L.F. fabricated the transmon qubits, while K.S.C., J.Z.B, and R.J.S. designed the experiment.
  • L.J., M.H.D., and L.F. provided theoretical support, and K.S.C. and R.J.S. wrote the manuscript with contributions from all authors.

S1. EXPERIMENTAL METHODS

The experiment uses coupled 3D cQED modules whose cavity, transmon, and readout modes are modeled with Kerr and dispersive Hamiltonian terms. The model includes relevant self-, dispersive, cross-, and direct-interaction parameters while neglecting specified small terms.

  • S1. EXPERIMENTAL METHODS: Each module contains a 3D cavity data qubit, a transmon qubit, and a Purcell-filtered readout resonator.The device uses three 3D cavities, two transmons, and two Purcell-filtered readout resonators.
  • S1. EXPERIMENTAL METHODS: The nine modes are divided into nearly-linear harmonic oscillators and anharmonic bosonic transmon modes, with physical parameters transformed into a dispersive Hamiltonian model.Finite-element simulation and black-box quantization provide the model parameters.
  • S1. EXPERIMENTAL METHODS: The Kerr oscillator Hamiltonian describes an anharmonic oscillator using resonance frequency ω_a and self-Kerr K_a.
  • S1. EXPERIMENTAL METHODS: The dispersive Hamiltonian describes interactions between modes through χ_ab and nonlinear dispersive interaction χ′_ab terms.With the transmon operated as a two-level system, the additional nonlinear term is ignored unless explicitly retained.
  • S1. EXPERIMENTAL METHODS: The module Hamiltonian includes cavity, transmon, and readout-resonator frequencies, self-Kerrs, dispersive interactions, and cross-Kerr χ_cr.The readout-resonator self-Kerr and cavity–readout nonlinear interaction are neglected as small perturbations.
  • S1. EXPERIMENTAL METHODS: The communication subsystem Hamiltonian includes two transmons, a bus cavity, bus–transmon interactions, and direct transmon coupling χ_q1q2.Nonlinear interaction terms do not contribute in the transmons’ two-level subspace.

C. Validity of module approximation

The authors assess whether locally coupled modules can be treated as separate by examining residual interactions. They conclude that the data qubits are effectively non-interacting for this experiment, while future remote modules would remove these residual interactions.

  • C. Validity of module approximation: The modular approximation is tested because residual interactions should be vanishingly small between separately housed data qubits.
  • C. Validity of module approximation: The dominant residual interaction is direct transmon coupling χ_q1q2, arising primarily through mutual interaction with the cavity mode.Its perturbative effect is reduced by its smaller magnitude, the antisymmetric Bell-pair choice, and insensitive local operations.
  • C. Validity of module approximation: The data-qubit–bus cross-Kerr interaction is bounded below a few kHz and estimated near 1 kHz, but it is not directly involved throughout the gate protocol.
  • C. Validity of module approximation: The data-qubit direct coupling χ_c1c2 has no statistically significant measured value, with simulations roughly predicting couplings of 1−10.
  • C. Validity of module approximation: Future implementations with remote logical-qubit modules would completely eliminate these residual interactions.

D. Experimental setup

The setup combines cryogenic microwave hardware with real-time quantum control, fast measurement, and feedback-based preparation. These capabilities support conditional operations while the experiment characterizes measurement independence and mitigates thermal populations.

  • D. Experimental setup: The device is mounted at 10−20 mK with magnetic shielding, radiation absorption, microwave attenuation, and filtering.
  • D. Experimental setup: Room-temperature IQ-mixer hardware generates shaped control pulses, while cryogenic and room-temperature amplification chains process readout signals.
  • D. Experimental setup: The control architecture generates pulses, samples measurements, and computes the next output in real time for shot-by-shot feedback or feedforward.
  • D. Experimental setup: Gaussian pulses with σ = 6 ns and DRAG corrections control transmon transitions and cavity displacements.
  • D. Experimental setup: 99.4% single-shot transmon assignment fidelity and approximately 1000 ns measurement-to-conditioned-operation latency enable classically conditional operations.The latency comprises a 600 ns measurement pulse, 200 ns cable delays, and 200 ns integration and estimation.
  • D. Experimental setup: Measurement crosstalk is assessed with simultaneous Rabi experiments, showing that each communication-qubit measurement is highly selective.
  • D. Experimental setup: Thermal populations near 10% for transmons and below 1% for cavities motivate a feedback-cooling sequence that resets transmons and empties cavities.

A. Implementation

The protocol initializes encoded data qubits, entangles communication qubits, applies local operations, and measures communication qubits to enact a teleported CNOT. Feedforward operations then make the data-qubit operation deterministic.

  • Initialization: The experiment initializes the system, encodes chosen states onto data qubits, and prepares communication qubits for the teleportation protocol.Encoding transfers a state prepared in communication qubits onto logical data-qubit states.
  • Entanglement: The communication-qubit Bell state is generated while the data qubits store their encoded information.The protocol uses an anti-symmetric Bell state for the communication qubits and tracks data-versus-communication states separately.
  • Initialization: System-wide reset uses measurement-based feedback and requires three consecutive ground-state measurements for both transmon qubits before accepting the reset.The transmon reset is also used as a subroutine for resetting the full system.
  • Local operations and measurement: A local CNOT is applied in each module before measuring the communication qubits in the Z and X bases.These measurements produce four uniformly distributed outcomes without revealing the data-qubit state.
  • Feedforward: The four measurement outcomes correspond to data-qubit states that differ by single-qubit operations but each implement a CNOT after correction.Feedforward operations transform the outcome-dependent states into the desired operation.

A. Data qubit encodings

The experiment uses binomial encoding to protect cavity-stored quantum information from single-photon loss and compares it with a simpler Fock encoding. Binomial encoding preserves amplitudes in a detectable error subspace.

  • Binomial encoding: Binomial encoding provides error correction against single-photon loss, the dominant error mechanism for a cavity quantum memory.The logical basis is designed so loss maps the state into identifiable error codewords.
  • Binomial encoding: A photon-loss event preserves the logical amplitudes while mapping the state into error codewords with odd photon-number parity.The error state is |ψE⟩ = α|E0⟩ + β|E1⟩, with |E0⟩ = |1⟩ and |E1⟩ = |3⟩.
  • Error correction: Photon-number parity measurements can detect the parity flip caused by single-photon loss and support a corrective unitary.The correction maps the error codewords back to the logical codewords in principle.
  • Fock encoding: Fock encoding uses the cavity’s lowest two energy levels but is not a logical encoding because it does not support quantum error correction.It provides an upper bound on teleported-gate performance with the current device.

1. Implementation by optimal control

The experiment uses GRAPE-based optimal control to implement universal operations between cavity data qubits and transmon communication qubits. Simulated pulses exceed 99% fidelity before modeled loss and decoherence reduce expected performance by 1–3%.

  • Optimal-control design: GRAPE designs time-dependent pulses that implement target unitaries or state transfers on selected cavity-transmon subspaces.The optimization uses the drift Hamiltonian and multiple control Hamiltonians to maximize fidelity.
  • Hardware constraints: Pulse optimization includes amplitude, derivative, and endpoint constraints to respect control-hardware limitations.These constraints limit drive strength, favor smooth low-bandwidth pulses, and require near-zero pulse endpoints.
  • Operation categories: The pulse set includes encoded-data single-qubit operations, data-communication entangling operations, and encoding or decoding pulses.Cavity and transmon are denoted by subscripts c and q, respectively.
  • Teleported-CNOT operations: The control-module operation is a cavity-controlled, transmon-target CNOT, while the target-module operation is transmon-controlled with a cavity target.The target-module operation flips the logical cavity state when the transmon is excited.
  • Encoding and decoding: Encoding pulses map arbitrary transmon states onto logical cavity states, and decoding pulses reverse this mapping for logical tomography.The transmon returns to its ground state after encoding and receives the decoded cavity state for measurement.
  • Fidelity: Simulated optimal-control fidelities exceed 99%, while Lindblad simulations including loss and decoherence are typically 1–3% lower.The ideal optimization excludes loss and decoherence; nonidealities are added afterward through master-equation simulation.

3. Experimental calibration

Experimental calibration focuses on accurate control-line characterization and the errors produced by optimal-control operations. Simulations identify codespace leakage as the dominant infidelity component and motivate leakage-detection circuits.

  • Calibration: Calibration of each module’s optimal-control pulses requires characterizing five control parameters, including cavity and transmon drive amplitudes.The calibration addresses the experiment’s control lines in addition to requiring an accurate drift Hamiltonian.
  • Future improvement: The authors identify leakage detection circuits as an outstanding question for efficiently heralding these errors.The dominant leakage error can be characterized by a single error syndrome.

C. Generation of communication qubit Bell pair

The communication-qubit Bell pair is generated with a refocused resonator-induced phase sequence designed to produce entanglement while suppressing residual interactions. The resulting Bell state has 97 ± 1% state fidelity.

  • Bell-pair generation: A resonator-induced phase gate uses an off-resonant shaped drive to generate a state-dependent entangling phase between the communication qubits.The drive is detuned from the bus resonance by approximately 20 MHz, and the entangling phase isolates the non-separable two-qubit contribution.
  • Bell-pair generation: A 300 ns refocused-RIP sequence sandwiches two RIP pulses around a π-pulse to echo away always-on dispersive interactions with the data cavities.The pulse shape is chosen to minimize residual photon population in the bus cavity.
  • Bell-pair characterization: 97 ± 1% state fidelity is achieved for the experimentally generated two-qubit Bell state.The reported error bar averages several experiments, while statistical errors are below 1%.
  • Bell-pair characterization: The local-operation infidelities for binomial and Fock encodings are characterized using predicted and experimental process results for encode-decode operations.These measurements provide the local-operation performance inputs used in the gate analysis.

D. Communication qubit measurement and reset

Communication-qubit measurements are combined with conditional reset so the qubits can be reused during the teleportation protocol. The protocol also tracks phase shifts and calibrates control and measurement operations needed for reliable execution.

  • Measurement and reset: After measurement, an excited communication qubit is conditionally flipped to the ground state, enabling reuse in subsequent protocol steps.The measurements must be non-destructive because the communication qubits serve multiple roles during teleportation.
  • Measurement and reset: Averaged reset infidelity is approximately 3%, primarily limited by decay during measurement and controller latency.Ground-state outcomes have infidelity below 1%, while excited-state outcomes show single-qubit infidelities of 2% and 4%.
  • Control calibration: Randomized benchmarking pulse sequences quantify the error per operation for the control pulses.The sequences compare the measured final state with the state expected after a correcting unitary.
  • Phase tracking: Logical and reference-frame phase shifts are distinguished because they have different effects on encoded cavity states.The dispersive interaction naturally generates a reference-frame phase shift when the communication qubit is excited.
  • Control calibration: RIP-gate tuning calibrates pulse amplitude, detuning, and duration through a Ramsey phase-style experiment.The refocused sequence explicitly includes the bus while entangling the communication qubits.
  • Measurement and reset: Conditional state tomography evaluates communication-qubit measurement and reset quality after applying outcome-dependent π-pulses.The initial state is an equal superposition of the four computational states.

B. Reference phases due to Bell state generation

Bell-pair generation induces reference-frame phase shifts on the data cavities, so these phases must be estimated and compensated before applying the local CNOT operations. Measurements and tomography verify the resulting phase behavior.

  • Bell-generation phases: Bell-pair generation requires phase adjustment because the data qubits already contain encoded information during communication-qubit entanglement.The spin-echo-like sequence yields estimated control and target phase shifts of 1.21 rad and 1.78 rad.
  • Bell-generation phases: A phase-extraction experiment verifies that the refocusing pulse disentangles the data and communication qubits by the end of Bell-state generation.The measured phase shift is 1.81 for both communication-qubit initial states and agrees with the simple estimate to approximately 1%.
  • Measurement phases: During communication-qubit measurement, the data cavity acquires no additional reference phase for |g⟩ and a phase θM = χTM for |e⟩.The phase depends on the total measurement duration.
  • Measurement phases: Wigner tomography extracts eight outcome-dependent phases, one for each communication-measurement outcome and data qubit.The extracted phases are stored by the controller and used as persistent reference-frame updates for later data-qubit operations.

D. Communication qubit measurement basis

The communication-qubit measurement basis controls a logical phase in the teleported operation, while process and error analyses quantify the resulting CNOT performance. The analysis also identifies encoding-dependent gate errors and a limitation in Bell-pair characterization.

  • Measurement basis: Changing the C2 measurement basis induces a logical phase, so the nominal X measurement angle must be calibrated against preceding single-qubit phases.The experiment varies the angle without feedforward and uses conditioned state tomography.
  • Measurement basis: The optimal measurement angle is extracted for implementing the teleported CNOT, and the basis choice may tune the particular teleported operation.The conditioned outputs remain maximally entangled over the tested angles.
  • Measurement basis: The two-qubit parity ⟨ZZ⟩ remains conserved across measurement angles, while transversal operators oscillate and state contrast stays constant.These observations indicate angle-dependent logical phase without changing the entanglement strength.
  • Process characterization: Process results are compiled for both binomial and Fock encodings, including communication-measurement-conditioned processes that expose the role of feedforward.Process fidelity includes encoding and decoding, while inferred gate fidelity removes their contribution relative to FE+D.
  • Local operations: The local CNOT operations use optimal-control pulses to map data-qubit encoding information onto communication qubits and enact the target-module logical operation.The control-module mapping is parity-based, while the target operation flips logical basis states when the communication qubit is excited.
  • Total infidelity: Bell-pair fidelity is estimated from characterization with unencoded data qubits, providing an upper bound for the encoded-data Bell pair.Encoded cavity photons make reliable Bell-pair tomography difficult, so the bare-state quantity is used in the error calculation.
  • Total infidelity: 16 ± 3% and 13 ± 2% are the expected total gate errors for binomial and Fock encodings, respectively.The inferred gate infidelities are 21 ± 2% and 13 ± 2%, consistent with the additive error model.

C. Contributions to infidelity

The teleported CNOT’s infidelity is dominated by finite transmon coherence, while tomography and simulations characterize additional limitations and reconstructed logical processes.

  • Contributions to infidelity: Finite transmon T2 and T1 contribute roughly 70% and 25% of total infidelity, respectively.Finite cavity lifetime contributes 4% and limits gate fidelity at 98%.
  • Contributions to infidelity: The error analysis finds no significant limitations from unexpected interactions or control errors at the measurement precision.
  • Contributions to infidelity: Without feedforward, the four measurement outcomes produce different processes and Bell states that combine into a completely mixed state.
  • Contributions to infidelity: With feedforward applied, every measurement outcome indicates the same process and the correct Bell state.

3. Data qubit state tomography

Data-qubit tomography indirectly characterizes logical states and processes by decoding cavity states onto communication qubits, while constrained reconstruction addresses noisy measurements and leakage.

  • Data qubit state tomography: Logical data-qubit tomography decodes states onto communication qubits, enabling trusted operations and measurements on the communication-qubit subspace.
  • Data qubit state tomography: Wigner tomography characterizes cavity states using displacements and a truncated Fock-basis reconstruction.
  • Data qubit state tomography: Maximum Likelihood Estimation constrains reconstructed states to remain physical when direct inversion produces unphysical density operators.
  • Data qubit state tomography: Quantum process tomography uses an overcomplete set of 36 logical two-qubit input states, followed by teleported CNOT application and decoding.
  • Data qubit state tomography: Codespace leakage errors are included rather than postselected away, although operations outside the encoded subspace are unconstrained.
Loading 1801.05283v1…