Quantum error correction protects logical quantum information against environmental decoherence by encoding logical qubits into entangled states of physical qubits. One of the most important near-term challenges in building a scalable quantum computer is to reach the break-even point, where logical quantum circuits on error-corrected qubits achieve higher fidelity than equivalent circuits on uncorrected physical qubits. Using Quantinuum's H2 trapped-ion quantum processor, we encode the Greenberger-Horne-Zeilinger (GHZ) state in four logical qubits with fidelity 99.5±0.15%≤F≤99.7±0.1% (after postselecting on over 98% of outcomes). Using the same quantum processor, we can prepare an uncorrected GHZ state on four physical qubits with fidelity 97.8±0.2%≤F≤98.7±0.2%. The logical qubits are encoded in a ⟦25,4,3⟧ Tanner-transformed long-range-enhanced surface code. Logical entangling gates are implemented using simple swap operations. Our results are a first step toward realizing fault-tolerant quantum computation with logical qubits encoded in geometrically nonlocal quantum low-density parity check codes.Quantum error correction protects logical quantum information against environmental decoherence by encoding logical qubits into entangled states of physical qubits. One of the most important near-term challenges in building a scalable quantum computer is to reach the break-even point, where logical quantum circuits on error-corrected qubits achieve higher fidelity than equivalent circuits on uncorrected physical qubits. Using Quantinuum's H2 trapped-ion quantum processor, we encode the Greenberger-Horne-Zeilinger (GHZ) state in four logical qubits with fidelity 99.5±0.15%≤F≤99.7±0.1% (after postselecting on over 98% of outcomes). Using the same quantum processor, we can prepare an uncorrected GHZ state on four physical qubits with fidelity 97.8±0.2%≤F≤98.7±0.2%. The logical qubits are encoded in a ⟦25,4,3⟧ Tanner-transformed long-range-enhanced surface code. Logical entangling gates are implemented using simple swap operations. Our results are a first step toward realizing fault-tolerant quantum computation with logical qubits encoded in geometrically nonlocal quantum low-density parity check codes.

arXiv:2406.02666Physical Review LettersOct 29, 202429 citations

Encoding and Node Choices in Transversal Fault-Tolerant Distributed Quantum Computations: An Initial Study

Seng W. Loke

We compare and study different Bivariate-Bicycle (BB) encodings and node choices for distributed quantum operations such as transversal non-local CNOTs. We observe that while some encodings have more physical qubits requiring more ebits for a distributed computation, reducing ebit consumption alone might not be the best criterion for selecting an encoding for the logical qubits, if the goal is to reduce the logical error rate of the distributed computation; for example, one should also consider encodings with a larger distance, which using more physical qubits can provide. We consider distributed, or non-local, CNOTs and computation of the global gate (GCZ) over distributed logical qubits as examples. The choice of encoding enables particular concurrent operations; e.g., we show that a transversal physical GCZ on a self-dual $[[120,8,12]] BB$ code can realize eight concurrent logical GCZ operations after accounting for the logical permutation induced by transversal Hadamard.We compare and study different Bivariate-Bicycle (BB) encodings and node choices for distributed quantum operations such as transversal non-local CNOTs. We observe that while some encodings have more physical qubits requiring more ebits for a distributed computation, reducing ebit consumption alone might not be the best criterion for selecting an encoding for the logical qubits, if the goal is to reduce the logical error rate of the distributed computation; for example, one should also consider encodings with a larger distance, which using more physical qubits can provide. We consider distributed, or non-local, CNOTs and computation of the global gate (GCZ) over distributed logical qubits as examples. The choice of encoding enables particular concurrent operations; e.g., we show that a transversal physical GCZ on a self-dual $[[120,8,12]] BB$ code can realize eight concurrent logical GCZ operations after accounting for the logical permutation induced by transversal Hadamard.

quant-pharXiv:2609.37210Sep 30, 20260 citations

Fault-tolerant encoding of a logical qubit

Norbert M. Linke

Bulletin of the American Physical SocietyJun 8, 20170 citations

Quantum Error Detection Without Using Ancilla Qubits

Nicolas J. Guerrero, David E. Weeks

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 $|0\rangle_L=|01\rangle$ and $|1\rangle_L=|10\rangle$ . If during the final measurement a $|11\rangle$ or $|00\rangle$ 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 $U_3$ and logical $C_x$ 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 $N\in\{2,3,4,5\}$ logical qubits and $Q\in\{1,2,3,4,5\}$ physical qubits per logical qubit. The results are compared relative to $Q=1$ with and without error detection and we find a significant improvement for $Q\in\{2,3,4\}$ .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 $|0\rangle_L=|01\rangle$ and $|1\rangle_L=|10\rangle$. If during the final measurement a $|11\rangle$ or $|00\rangle$ 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 $U_3$ and logical $C_x$ 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 $N\in\{2,3,4,5\}$ logical qubits and $Q\in\{1,2,3,4,5\}$ physical qubits per logical qubit. The results are compared relative to $Q=1$ with and without error detection and we find a significant improvement for $Q\in\{2,3,4\}$.

arXiv:2204.11114Apr 23, 20220 citations

Basic Operations with Qubits

Sergei Kurgalin, Sergei Borzunov

Texts in computer scienceJan 1, 20211 citation

[[120,8,12]] BB

code can realize eight concurrent logical GCZ operations after accounting for the logical permutation induced by transversal Hadamard.

|0\rangle_L=|01\rangle

and

|1\rangle_L=|10\rangle

. If during the final measurement a

|11\rangle

|00\rangle

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

U_3

and logical

C_x

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\in\{2,3,4,5\}

logical qubits and

Q\in\{1,2,3,4,5\}

physical qubits per logical qubit. The results are compared relative to

Q=1

with and without error detection and we find a significant improvement for

Q\in\{2,3,4\}