Quantum Error Detection Without Using Ancilla Qubits

Abstract

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 $|0{⟩}_L=|01⟩$ and $|1{⟩}_L=|10⟩$. If during the final measurement a $|11⟩$ or $|00⟩$ 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 $U_3$ 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 $Q\in \{2,3,4\}$.