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arXiv:1812.09278 · Physical Review A

Efficiencies of logical Bell measurements on Calderbank-Shor-Steane codes with static linear optics

Frank L. Schmidt, Peter van Loock

Published Jun 10, 2019• 18 citations
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Abstract

We show how the efficiency of a logical Bell measurement (BM) can be calculated for arbitrary Calderbank-Shor-Steane (CSS) codes with the experimentally important constraint of using only transversal static linear-optical BMs on the physical single-photon qubit level. For this purpose, we utilize the codes' description in terms of stabilizers in order to calculate general efficiencies for the loss-free case, but also for specific cases including photon loss. These efficiencies can be, for instance, used for obtaining transmission rates of all-optical quantum repeaters. In the loss-free case, we demonstrate that the important class of CSS codes with identical physical-qubit support for the two logical Pauli ($Z$ and $X$) operators can only achieve a logical BM efficiency of $\frac{1}{2}$ if one always employs the same ancilla-free static linear optical BMs on the physical level. We apply our methods to various CSS codes including two-dimensional planar color and planar surface codes. We then find that in many cases, the fixed use of the standard linear-optical BM for all physical BMs is suboptimal and performing linear-optical transformations before doing the standard linear-optical BM (still without any ancillary photons and without any feedforward) can increase the efficiency enormously. In fact, using this generalization in the no-loss (or sometimes also in the low-loss) case allows us, on the one hand, to improve the logical BM efficiency of quantum parity codes compared to previously known results and, on the other hand, it also enables us to enhance the efficiency of two-dimensional planar color codes, whose efficiency is otherwise subject to the above $\frac{1}{2}$ bound, to arbitrarily close to unity.

DOI
10.1103/physreva.99.062308

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We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical controlled NOT operator for this code. We also show how to convert this code into a nontrivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a nondegenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver's half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender's side. We discuss the advantages and disadvantages for each of the two codes.

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We show how the efficiency of a logical Bell measurement (BM) can be calculated for arbitrary Calderbank-Shor-Steane (CSS) codes with the experimentally important constraint of using only transversal static linear-optical BMs on the physical single-photon qubit level. For this purpose, we utilize the codes' description in terms of stabilizers in order to calculate general efficiencies for the loss-free case, but also for specific cases including photon loss. These efficiencies can be, for instance, used for obtaining transmission rates of all-optical quantum repeaters. In the loss-free case, we demonstrate that the important class of CSS codes with identical physical-qubit support for the two logical Pauli (ZZZ and XXX) operators can only achieve a logical BM efficiency of 12\frac{1}{2}21​ if one always employs the same ancilla-free static linear optical BMs on the physical level. We apply our methods to various CSS codes including two-dimensional planar color and planar surface codes. We then find that in many cases, the fixed use of the standard linear-optical BM for all physical BMs is suboptimal and performing linear-optical transformations before doing the standard linear-optical BM (still without any ancillary photons and without any feedforward) can increase the efficiency enormously. In fact, using this generalization in the no-loss (or sometimes also in the low-loss) case allows us, on the one hand, to improve the logical BM efficiency of quantum parity codes compared to previously known results and, on the other hand, it also enables us to enhance the efficiency of two-dimensional planar color codes, whose efficiency is otherwise subject to the above 12\frac{1}{2}21​ bound, to arbitrarily close to unity.
Bell measurements (BMs) are ubiquitous in quantum information and technology. They are basic elements for quantum commmunication, computation, and error correction. In particular, when performed on logical qubits encoded in physical photonic qubits, they allow for a read-out of stabilizer syndrome information to enhance loss tolerance in qubit-state transmission and fusion. However, even in an ideal setting without photon loss, BMs cannot be done perfectly based on the simplest experimental toolbox of linear optics. Here we demonstrate that any logical BM on stabilizer codes can always be mapped onto a single physical BM perfomed on any qubit pair from the two codes. As a necessary condition for the success of a logical BM, this provides a general upper bound on its success probability, especially ruling out the possibility that the stabilizer information obtainable from only partially succeeding, physical linear-optics BMs could be combined into the full logical stabilizer information. We formulate sufficient criteria to find schemes for which a single successful BM on the physical level will always allow to obtain the full logical information by suitably adapting the subsequent physical measurements. Our approach based on stabilizer group theory is generally applicable to any stabilizer code, which we demonstrate for quantum parity, five-qubit, standard and rotated planar surface, tree, and seven-qubit Steane codes. Our schemes attain the general upper bound for all these codes, while this bound had previously only been reached for the quantum parity code.
In this paper, we demonstrate a direct correspondence between phase-free ZX diagrams, a graphical notation for representing and manipulating a certain class of linear maps on qubits, and Calderbank-Shor-Steane (CSS) codes, a large family of quantum error correcting codes constructed from classical codes, including for example the Steane code, surface codes, and colour codes. The stabilisers of a CSS code have an especially nice structure arising from a pair of orthogonal F2\mathbb F_2F2​-linear subspaces, or in the case of maximal CSS codes, a single subspace and its orthocomplement. On the other hand, phase-free ZX diagrams can always be efficiently reduced to a normal form given by the basis elements of an F2\mathbb F_2F2​-linear subspace. Here, we will show that these two ways of describing a quantum state by an F2\mathbb F_2F2​-linear subspace SSS are in fact the same. Namely, the maximal CSS code generated by SSS fixes the quantum state whose ZX normal form is also given by SSS. This insight gives us an immediate translation from stabilisers of a maximal CSS code into a ZX diagram describing its associated state. We show that we can extend this translation to stabilisers and logical operators of any (possibly non-maximal) CSS code by "bending wires". To demonstrate the utility of this translation, we give a simple picture of the surface code and a fully graphical derivation of the action of physical lattice surgery operations on the space of logical qubits, completing the ZX presentation of lattice surgery initiated by de Beudrap and Horsman.
Realizing Logical Diagonal Gates via Transversal Physical ZZZ-Rotations in CSS Codes
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1C_2 \subseteq C_1C2​⊆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)(C_1, C_2)(C1​,C2​) whose resulting CSS codes realize a target logical diagonal gate via transversal physical ZZZ-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit ZZZ-rotations and multi-qubit controlled ZZZ-rotations via transversal physical ZZZ-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an [[n′,k′]][[n',k']][[n′,k′]] CSS code Q′Q'Q′ and a target logical ZZZ-rotation (single-qubit or multi-controlled) ULU_LUL​, and extends Q′Q'Q′ by systematically appending n′′n''n′′ physical qubits to obtain an [[n,k]][[n,k]][[n,k]] CSS code QQQ with n=n′+n′′n = n'+n''n=n′+n′′ and k=k′k=k'k=k′. The target logical gate ULU_LUL​ is realized in QQQ by applying a well-chosen physical transversal ZZZ-rotation to the n′′n''n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical ZZZ-rotations in the input code Q′Q'Q′ remains transversally realizable in the extended code QQQ. The CSS code QQQ 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′Q'Q′ to obtain a CSS code QQQ that supports fault-tolerant implementations of multiple desired logical ZZZ-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 ZZZ-rotations.
Shor and Steane ancilla are two well-known methods for fault-tolerant logical measurements, which are successful on small codes and their concatenations. On large quantum low-density-parity-check (LDPC) codes, however, Shor and Steane measurements have impractical time and space overhead respectively. In this work, we widen the choice of ancilla codes by unifying Shor and Steane measurements into a single framework, called homomorphic measurements. For any Calderbank-Shor-Steane (CSS) code with the appropriate ancilla code, one can avoid repetitive measurements or complicated ancilla state preparation procedures such as distillation, which overcomes the difficulties of both Shor and Steane methods. As an example, we utilize the theory of covering spaces to construct homomorphic measurement protocols for arbitrary XXX- or ZZZ-type logical Pauli operators on surface codes in general, including the toric code and hyperbolic surface codes. Conventional surface code decoders, such as minimum-weight perfect matching, can be directly applied to our constructions.
This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal ZZZ-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal TTT realizes the logical transversal S†S^{\dagger}S†; (b) show that the condition C2∗C1⊆C1⊥C_2 \ast C_1 \subseteq C_1^{\perp}C2​∗C1​⊆C1⊥​ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal TTT implements the logical identity and the logical transversal TTT, respectively.
We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical controlled NOT operator for this code. We also show how to convert this code into a nontrivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a nondegenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver's half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender's side. We discuss the advantages and disadvantages for each of the two codes.