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arXiv:quant-ph/0408064 · Physical Review A

High-fidelity Z -measurement error encoding of optical qubits

Jeremy L. O’Brien, Geoff J. Pryde, Andrew G. White, Timothy C. Ralph

Published Jun 9, 2005• 25 citations
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Abstract

We demonstrate a quantum error correction scheme that protects against accidental measurement, using a parity encoding where the logical state of a single qubit is encoded into two physical qubits using a nondeterministic photonic controlled-NOT gate. For the single qubit input states $\ensuremath{\mid}0⟩$, $\ensuremath{\mid}1⟩$, $\ensuremath{\mid}0⟩\ifmmode\pm\else\textpm\fi{}\ensuremath{\mid}1⟩$, and $\ensuremath{\mid}0⟩\ifmmode\pm\else\textpm\fi{}i\ensuremath{\mid}1⟩$ our encoder produces the appropriate two-qubit encoded state with an average fidelity of $0.88\ifmmode\pm\else\textpm\fi{}0.03$ and the single qubit decoded states have an average fidelity of $0.93\ifmmode\pm\else\textpm\fi{}0.05$ with the original state. We are able to decode the two-qubit state (up to a bit flip) by performing a measurement on one of the qubits in the logical basis; we find that the 64 one-qubit decoded states arising from 16 real and imaginary single-qubit superposition inputs have an average fidelity of $0.96\ifmmode\pm\else\textpm\fi{}0.03$.

DOI
10.1103/physreva.71.060303

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Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD trapped-ion quantum computer to encode a single logical qubit using the [[7,1,3]][[7,1,3]][[7,1,3]] color code, first proposed by Steane~\cite{steane1996error}. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of 1.7(6)×10−31.7(6) \times 10^{-3}1.7(6)×10−3, compared to the average physical SPAM error 2.4(8)×10−32.4(8) \times 10^{-3}2.4(8)×10−3 of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a logical magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudo-threshold.
Quantum information can be protected from noise and decoherence by encoding it into a decoherence-free subspace of the full Hilbert space. In this approach, it is important to devise implementations of quantum logic gates acting on the encoded logical qubits. Here, we experimentally investigate the implementation of an entangling linear optical controlled-ZZZ gate on two logical qubits encoded into pairs of physical qubits, which protects the quantum information from collective dephasing. This decoherence-resilient two-qubit gate on logical qubits is realized using a suitable four-qubit quantum gate on the physical qubits, which may help to protect the encoded quantum information from systematic errors arising due to imperfect operations on single logical qubits. We characterize the gate operation on input superposition states and demonstrate its ability to protect quantum coherence even under complete collective dephasing.
Long-distance quantum communication with high fidelity is the main obstacle to creating a quantum network. A scheme to recover the loss of one photon with a three-to-eight qubit encoding is presented. The qubit loss is first detected using a quantum nondemolition measurement and is then transformed into a standard qubit error by inserting a new photon. The state of qubits is reconstructed via a sequence of one projective measurement, two single-qubit gates, and seven controlled-NOT operations. No ancillary qubits are required. Finally, the cost of resources is analyzed.
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, midcircuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a 10-qubit quantum charge-coupled device trapped-ion quantum computer to encode a single logical qubit using the [[7,1,3]] color code, first proposed by Steane [Phys. Rev. Lett. 77, 793 (1996)PRLTAO0031-900710.1103/PhysRevLett.77.793]. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical state preparation and measurement (SPAM) error of 1.7(2)×10^{-3}, compared to the average physical SPAM error 2.4(4)×10^{-3} of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a T[over ¯]|+⟩_{L} magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudothreshold.
In this paper, we describe and experimentally demonstrate an error detection scheme that does not employ ancilla qubits or mid-circuit measurements. This is achieved by expanding the Hilbert space where a single logical qubit is encoded using several physical qubits. For example, one possible two qubit encoding identifies ∣0⟩L=∣01⟩|0\rangle_L=|01\rangle∣0⟩L​=∣01⟩ and ∣1⟩L=∣10⟩|1\rangle_L=|10\rangle∣1⟩L​=∣10⟩. If during the final measurement a ∣11⟩|11\rangle∣11⟩ or ∣00⟩|00\rangle∣00⟩ is observed an error is declared and the run is not included in subsequent analysis. We provide codewords for a simple bit-flip encoding, a way to encode the states, a way to implement logical U3U_3U3​ and logical CxC_xCx​ gates, and a description of which errors can be detected. We then run Greenberger-Horne-Zeilinger circuits on the transmon based IBM quantum computers, with an input space of N∈{2,3,4,5}N\in\{2,3,4,5\}N∈{2,3,4,5} logical qubits and Q∈{1,2,3,4,5}Q\in\{1,2,3,4,5\}Q∈{1,2,3,4,5} physical qubits per logical qubit. The results are compared relative to Q=1Q=1Q=1 with and without error detection and we find a significant improvement for Q∈{2,3,4}Q\in\{2,3,4\}Q∈{2,3,4}.