We consider 2nd generation (2G) Quantum Repeaters (QR) for creating long-distance entanglement in quantum networks. Combining a distance-dependent depolarizing error model with the non-local Bell-state purification procedure required by 2G QRs leads to an error model consisting of correlated and biased errors. To correct correlated errors, non-symmetric CSS codes with joint decoding between stations can be used. The dominating errors are biased, such that different repeater stations suffer from different types of errors. To mitigate this, different quantum codes can be used at the stations, optimized for the specific error model of the station. To comply with the 2G QR procedure, the codes used in neighboring stations must allow for the transversal implementation of non-local logical CNOT gates across the two stations, or alternatively, non-local CZ gates combined with logical Hadamards. We provide a complete characterization of pairs of Calderbank-Steane-Shor (CSS) codes that allow CNOT- or CZ-transversality, and examine an explicit family of mirrored CSS codes allowing CZ-transversality. We verify the Hadamard gate transversality using our framework and show the importance of the mapping matrix. Also, we conclude that using different QECCs can not achieve transversal universal computation with Clifford +
T-gate set. Finally, we study the entanglement generation rate (EGR) in 2G QRs with limited quantum memory, minimizing the number of intermediate stations for a given fidelity and EGR. By simulation we observe that non-symmetric and mirrored structure QECCs outperform the conventional approach of using symmetric CSS codes at the repeater stations.
In this paper, we investigate the transversality of pairs of CSS codes and their use in the second generation of quantum repeaters (QR)s. We show that different stations of quantum link can experience different errors. Considering this fact, we suggest to use different CSS codes in different stations. We also suggest to use
[[n,k]] codes with
k>1 as they are more efficient then codes with
k=1. We establish sufficient and necessary conditions for a pair of CSS codes to be non-local CNOT-transversal. We show that in contrast to the well known CNOT transversality which states that two CSS codes should be the same, less restrictive constraints are needed. Next, we establish sufficient and necessary conditions for a code pair to be CZ-transversal.
For achieving long-distance quantum communication, Quantum Repeaters (QRs) have to be used, with commu-nication range and reliability increased by using intermediate stations. The physical requirements of second generation QRs may be achievable in the near future. They use Quantum Error Correction Codes (QECC) to protect logical qubits against environmental interaction using physical redundancy. In this work, we study the types of errors that can corrupt quantum codewords in intermediate stations. Our studies show that the errors are distance-dependent, and also consist of correlated errors and biased errors. To mitigate this error model, we use non-symmetric CSS codes as well as mirrored structure coding. We show that using non-symmetric CSS codes results in better performance. Also, we prove the logical CZ gate transversality of the mirrored structure coding. The effectiveness of the proposed methods is verified by numerical simulations.
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes
C2⊆C1, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs
(C1,C2) whose resulting CSS codes realize a target logical diagonal gate via transversal physical
Z-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit
Z-rotations and multi-qubit controlled
Z-rotations via transversal physical
Z-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an
[[n′,k′]] CSS code
Q′ and a target logical
Z-rotation (single-qubit or multi-controlled)
UL, and extends
Q′ by systematically appending
n′′ physical qubits to obtain an
[[n,k]] CSS code
Q with
n=n′+n′′ and
k=k′. The target logical gate
UL is realized in
Q by applying a well-chosen physical transversal
Z-rotation to the
n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical
Z-rotations in the input code
Q′ remains transversally realizable in the extended code
Q. The CSS code
Q may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code
Q′ to obtain a CSS code
Q that supports fault-tolerant implementations of multiple desired logical
Z-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled
Z-rotations.
Abstract Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.
The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to
n qubits with T-count
3n−3. Our main result is the construction of a novel qutrit
[[6,2,2]] quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a
[[48,2,4]] code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.