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Experimental demonstration of fault-tolerant state preparation with superconducting qubits
Maika Takita, Andrew W. Cross, A. D. Córcoles, Jerry M. Chow, Jay M. Gambetta
TL;DR
The paper examines how CNOT-sequence-dependent phase errors affect encoded-state preparation and logical-state evolution. It compares two CNOT implementations, models error insertion, and observes phase sensitivity and free-evolution coherence loss consistent with static ZZ interactions.
Problem
The study addresses how systematic phase errors affect preparation of different encoded codewords and how encoded logical observables evolve freely.
Method
The authors compare two-pulse and four-pulse CNOT sequences, use reconstructed states to assess preparation, and model error insertion through acceptance and conditional logical-error coefficients.
Results
The four-pulse sequence effectively cancels the phase error, while encoded |¯+¯+⟩ evolution shows rapid decay and coherent oscillation consistent with static ZZ terms; fitted parameters are ˜p0 = 0.108 and ˜p1 = 0.043.
Takeaways & Limitations
CNOT echoing can reduce the phase-error sensitivity of state preparation, while static ZZ interactions constrain the coherence of encoded logical observables during free evolution.
Abstract
from arXiv · showhide
Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit superconducting transmon device and demonstrate fault-tolerant state preparation. We characterize the resulting codewords through quantum process tomography and study the free evolution of the logical observables. Our results are consistent with fault-tolerant state preparation in a protected qubit subspace.
DEVICE PARAMETERS
The device-characterization tables report qubit, readout, coherence, static ZZ, and randomized-benchmarking parameters used to assess the hardware.
- QUBIT AND READOUT CHARACTERIZATION: Table S1 reports qubit transitions, relaxation and coherence times, readout resonator frequencies, readout assignment errors, and single-qubit randomized-benchmarking error per gate.Single-qubit gates are 85 ns long, and qubit anharmonicities are around 330 MHz.
- STATIC ZZ CHARACTERIZATION: Table S2 reports static ZZ strengths measured with pi-Hahn echo experiments for each qubit pair.The table notes εzz < 5 kHz.
FOUR QUBIT CODE
The supplementary figure presents an encoder for two logical qubits and four syndrome-bit configurations in the four-qubit code.
- FOUR-QUBIT CODE: The encoder maps logical qubits L1 and L2 into the four-qubit code with syndrome-bit values sz and sx.The encoder is used implicitly in state-tomography analysis rather than physically implemented.
FAULT-TOLERANT STATE PREPARATION
The study compares fault-tolerant state-preparation behavior across codewords, CNOT decompositions, and systematic phase-error sensitivities in the four-qubit code.
- FAULT-TOLERANT STATE PREPARATION: State-tomography comparisons use FPCX and TPCX CNOT sequences for prepared logical states in both Z and X bases.The two sequences’ pulse decompositions are shown in Fig. S4, with two-qubit error-per-gate results compared in Table S4.
- FAULT-TOLERANT STATE PREPARATION: TPCX preparations show codeword-dependent sensitivity to systematic phase errors, with |¯0p¯0g⟩ and |¯±⟩ states more sensitive than the other logical-basis states.The |¯+g ¯+p⟩ state prepared with TPCX has low acceptance probability and is dominated by |¯+g ¯+p, ˜0˜1⟩.
- FAULT-TOLERANT STATE PREPARATION: AC Stark shifts may accumulate as relative phase 4θ on |¯0p¯0g⟩, while the FPCX sequence effectively cancels this error.The passage states that the error effectively cancels on the other listed states.
- FAULT-TOLERANT STATE PREPARATION: Table S3 reports acceptance probability and protected- and gauge-qubit preparation errors conditioned on the initial state being in the codespace.The measurements compare two different CNOT gate sequences.
ERROR INSERTION
The error-insertion analysis fits acceptance and conditional logical-error data using parameters for different insertion locations and qubits.
- ERROR INSERTION: The fitted phase-offset parameters are δA = −0.1369, δB = −0.2291, and δC = 0.0278.δA and δC are determined from acceptance data, whereas δB is fitted from conditional logical-error data.
- ERROR INSERTION: The model’s acceptance-probability coefficients are expressed in terms of readout-error parameters p0 and p1.The supplied equations give separate coefficient expressions for the insertion locations, including aB.
- ERROR INSERTION: The conditional logical-error coefficients are likewise expressed as functions of p0 and p1.The supplied expression defines coefficients used for conditional logical-error probabilities.
- ERROR INSERTION: At location A, the L2 gauge-qubit logical-error probability equals the L1 protected-qubit logical-error probability.This equality is stated for the location-A error-insertion analysis.
ENCODED |¯+¯+⟩LIFETIME
The encoded |¯+¯+⟩ state undergoes rapid decay and coherent oscillation, consistent with free evolution from static ZZ interactions. Echo sequences slow the decay, while preparation-circuit Pauli placement affects idle-phase accumulation.
- ENCODED |¯+¯+⟩LIFETIME: Rapid decay and coherent oscillation occur for both encoded qubits in the |¯+¯+⟩ state.The behavior is consistent with free evolution under static ZZ terms in the Hamiltonian.
- ENCODED |¯+¯+⟩LIFETIME: A static ZZ strength η of about 50 kHz gives a tπ timescale of 10µs, consistent with the observed loss of coherence.
- ENCODED |¯+¯+⟩LIFETIME: Without echo pulses, the codeword decays to 1/2 in about 4µs and oscillates for about 20µs.Echo sequences produce decay rates comparable to those of the physical qubits.
- ENCODED |¯+¯+⟩LIFETIME: Preparation circuits differ in Pauli-operator placement, which affects phase accumulation while qubits are idle.