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arXiv:2207.04996

Construction of non-CSS quantum codes using measurements on cluster states

Swayangprabha Shaw, Gupta, Harsh, Shahid Mehraj Shah, Ankur Raina

Published Jul 11, 2022• 1 citation
arXivHTMLPDF

Abstract

The Measurement-based quantum computation provides an alternate model for quantum computation compared to the well-known gate-based model. It uses qubits prepared in a specific entangled state followed by single-qubit measurements. The stabilizers of cluster states are well defined because of their graph structure. We exploit this graph structure extensively to design non-CSS codes using measurement in a specific basis on the cluster state. % We aim to construct $[[n,1]]$ non-CSS code from a $(n+1)$ qubit cluster state. The procedure is general and can be used specifically as an encoding technique to design any non-CSS codes with one logical qubit. We show there exists a $(n+1)$ qubit cluster state which upon measurement gives the desired $[[n,1]]$ code.

DOI
10.48550/arxiv.2207.04996

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The Measurement-based quantum computation provides an alternate model for quantum computation compared to the well-known gate-based model. It uses qubits prepared in a specific entangled state followed by single-qubit measurements. The stabilizers of cluster states are well defined because of their graph structure. We exploit this graph structure extensively to design non-CSS codes using measurement in a specific basis on the cluster state. % We aim to construct [[n,1]][[n,1]][[n,1]] non-CSS code from a (n+1)(n+1)(n+1) qubit cluster state. The procedure is general and can be used specifically as an encoding technique to design any non-CSS codes with one logical qubit. We show there exists a (n+1)(n+1)(n+1) qubit cluster state which upon measurement gives the desired [[n,1]][[n,1]][[n,1]] code.
We generalize the unified framework in [Phys. Rev. A 113, 022438 (2026)] to accommodate arbitrary stabilizer codes (referred as non-CSS for emphasis). The generalization has immediate consequences in areas including logical measurement, quantum weight reduction and Euclidean embedding. For example, currently CSS (pure X,ZX,ZX,Z-type) logical measurements are well-understood based on the (height-1) cone, while non-CSS (e.g., YYY-type) logical measurements are ad hoc (e.g., rely on local Clifford transform). Similarly, quantum weight reduction and optimal Euclidean embeddings only exist for CSS input codes. Here, we show that our generalized framework addresses this issue, and thus many constructions, including qLDPC surgery, Layer Codes and quantum weight reduction generalize to non-CSS codes in a relatively straightforward fashion. The previously mentioned examples are derived in detail for clarity.
Quantum communication and computation is an emerging field in quantum mechanics. There are many promising applications including secure communications, simulation of complex quantum systems or quantum algorithms which perform faster than classical methods. The main barrier to realising these possibilities is coping with the inevitable errors induced by noise and experimental imperfections. Quantum error correction is a process which allows for the storage and retrieval of encoded qubits with high fidelity even in an imperfect environment. Error correction operates by encoding qubit states into a subspace of a larger system of qubits. The presence of errors is detected by performing stabilizer measurements. These measurements do not act on the encoding subspace and the measurement results, or syndrome, can be used to infer errors. The decoding problem is to use syndrome information to find the most likely errors which have occurred. This is performed using classical computing resources. Once the errors have been inferred operations can be applied to the system to counteract their effect. While many quantum error correcting codes are known not all are suitable in a practical setting. When choosing a potential code we must consider the following. Does the construction use simple interactions that can be physically realised? How high is the physical qubit to encoded qubit overhead? Does an efficient decoder exist? Is the encoding scheme scalable and do the error correcting properties improve with scaling? All of these properties are desirable for a practical scheme, though some compromise may be required. In this thesis I show how cluster states can be used as a resource for implementing CSS quantum codes. Additionally I show how these cluster codes can be linked together in a layered fashion to create foliated codes. These foliated codes can be used with any constituent CSS code as a basis for fault tolerant error correction. This process is an extension of Raussendorf’s toric code measurement based processing scheme. I first generalise this measurement based scheme to all CSS quantum codes, which is presented in chapter 3. I then present a novel decoding scheme for foliated turbo codes, chapter 4, and present the decoding performance of some turbo code families in chapter 5. These turbo codes are a class of finite rate codes whose overhead and complexity are flexible. Chapter 6 presents the decoding performance of foliated bicycle codes. The low overhead of physical resources to logical qubits of both these code families make these foliated schemes an attractive alternative to the toric code when considering practical measurement based computing designs with large numbers of encoded qubits.
Topological measurement-based quantum computation (MBQC) enables one to carry out universal fault-tolerant quantum computation via single-qubit measurements with a family of large entangled states called cluster states as resources. Raussendorf's three-dimensional cluster states (RTCSs) based on the surface codes are mainly considered for topological MBQC. In such schemes, however, the logical Hadamard, phase (Z1/2)({Z}^{1/2})(Z1/2), and TTT (Z1/4)({Z}^{1/4})(Z1/4) gates which are essential for building up arbitrary logical gates are not implemented natively without using state distillation or lattice dislocations, to the best of our knowledge. In particular, state distillation generally consumes many ancillary logical qubits; thus it is a severe obstacle against practical quantum computing. To solve this problem, we suggest an MBQC scheme via a family of cluster states called color-code-based cluster states (CCCSs) based on the two-dimensional color codes instead of the surface codes. We define logical qubits, construct elementary logical gates, and describe error correction schemes. We show that all logical Clifford gates generated by the cnot, Hadamard, and phase gates can be implemented natively in a fault-tolerant manner, although the TTT gate still requires state distillation to be fault-tolerant. The error thresholds of MBQC via CCCSs for logical-ZZZ errors are calculated to be 2.7%--2.8%, which are comparable to the values for RTCSs, assuming a simple error model where physical qubits have nontrivial errors independently with the same probability. We analyze and compare the resource overheads of both the schemes. In particular, we show that the number of physical qubits required for implementing a phase gate with CCCSs is at least about 26 times smaller than with RTCSs using state distillation, for the same code distance.
nnn-qubit,mmm-uniform states are[[n, 0, m+1]][[n,\ 0,\ m+1]][[n, 0, m+1]]nonde-generate quantum error correcting codes (QECC). The codeword stabilized (CWS) codes use [n,k,d≥m+1n, k, d\geq m+1n,k,d≥m+1] classical codes andmmm-uniform graph states to construct[[n, k, m+1]][[n,\ k,\ m+1]][[n, k, m+1]]QECCs. We provide a set of conditions on the classical codes to obtain CWS codes, which can be encoded in a proposed measurement-based approach. We arrive at an explicit construction of a CWS code when the startingmmm-uniform graph state is a cluster state of qubits on a one-dimensional lattice. The proposed encoding protocol admits sequential encoding of logical qubits, making it suitable for quantum memory applications.
Although it is well-known that all quantum states can be produced by single-qubit measurements on the cluster states, it is not a simple task to explicitly find which measurement patterns on the cluster states can generate different quantum states. In this paper, we introduce a general scheme to find measurement patterns corresponding to Calderbank--Shor--Steane (CSS) topological states containing Kitaev's toric code states and color code states on different lattices and in different dimensions. Furthermore, we find a measurement pattern for generating non-Abelian anyons where measurement-induced defects on a toric code state play the role of Ising anyons. We also support our scheme by a graphical notation where, by following a few simple graphical transformations, one will be able to convert a CSS topological state to a cluster state. Our scheme can also be used for experimental realization of anyons on cluster states.
Modularization promises to break down the design and implementation complexity of large scale quantum processors into smaller manageable subtasks. In this approach, quantum channels, realized for instance through physical rerouting of qubits or quantum teleportation, connect multiple modules. Each of those modules hosts a subset of qubits, e.g. multiple logical qubits, and provides quantum operations on them. In this work, we implement for the first time all logical operations required for modular fault-tolerant universal quantum computing with a non-Calderbank-Shor-Steane (CSS) code, the perfect [[5,1,3]][[5, 1, 3]][[5,1,3]] code, on a trapped-ion quantum computer. This code is the smallest quantum error-correcting (QEC) code capable of correcting any single-qubit error, making it a compact alternative to larger CSS codes. We demonstrate logical state teleportation and a full suite of fault-tolerant operations required for universal logical control, including logical state preparation, QEC with real-time feedback, logical measurements, magic-state preparation, logical entangling operations, and magic-state injection. Moreover, we characterize the logical spectator error picked up by idling logical qubits during quantum operations on distinct qubit registers and demonstrate a logical Pauli quantum process tomography that minimizes required sampling resources for logical tomography.