Universal fault-tolerant measurement-based quantum computation Benjamin J. Brown and Sam Roberts Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia Certain physical systems that one might consider for fault-tolerant quantum computing where qubits do not readily interact, for instance photons, are better suited for measurement-based quantum-computational protocols. Here we propose a measurement-based model for universal quan- tum computation that simulates the braiding and fusion of Majorana modes. To derive our model we develop a general framework that maps any scheme of fault-tolerant quantum computation with stabilizer codes into the measurement-based picture. As such, our framework gives an explicit way of producing fault-tolerant models of universal quantum computation with linear optics using protocols developed using the stabilizer formalism. Given the remarkable fault-tolerant properties that Majorana modes promise, the main example we present offers a robust and resource efficient proposal for photonic quantum computation. I. INTRODUCTION The potential savings available by adopting our robust model of quantum computation are twofold. Firstly, we achieve the Clifford operations by simulating the braids Considerable experimental effort [1{18] is being dedi- and fusion measurements of Majorana zero modes with cated to the realization of linear optical quantum com- the topological cluster-state model. We find these oper- putation [19, 20] due to the extensive coherence times ations by mapping them from analogous operations us- that photonic qubits promise. To support this endeavor, ing stabilizer models that are known to be resource ef- we must design robust models of fault-tolerant quantum ficient [45{48] compared with the original proposal [44]. computation that minimize the high resource cost that Furthermore, unlike our model, the original proposal re- is demanded by their implementation. Unlike the other lied on resource-intensive distillation methods to com- approaches to realize fault-tolerant quantum computa- plete the set of Clifford operations [49]. We expect addi- tion [21{28], qubits encoded with photons fly through tional reductions in resource demands by circumventing space at the speed of light, and do not readily interact the need to perform any distillation operations to per- with other photons. As such, we need different theoreti- form Clifford gates. cal models to describe fault-tolerant quantum computa- We derive our results by developing a framework that tion with photons. allows us to map any protocol for quantum compu- Measurement-based quantum computation [29{33] tation with stabilizer codes [50] onto a measurement- provides a natural language to describe computational based model. Our mapping is therefore important in its operations with flying qubits. In this picture we initial- own right as it enables us to import all of the devel- ize a specific many-body entangled resource state which opments made for fault-tolerant quantum computation is commonly known as a cluster state. We then make with static qubits using the stabilizer formalism in the single-qubit measurements on its qubits to realize com- past decades [51{53] into a language that is better suited putational operations. The performed operation is deter- for linear-optical setups. Specifically, we show how to mined by the choice of measurements we make, and the encode an arbitrary stabilizer state with a cluster-state entangled resource we initially produced. model. We also show how to choose the cluster state and In the present work, we provide a fault-tolerant im- the measurement pattern such that we implement logical plementation of the Clifford operations with an adapta- operations on the encoded stabilizer code. We accom- tion of the topological cluster-state model [34]. If sup- plish logical operations using a specially chosen quantum arXiv:1811.11780v2 [quant-ph] 25 Aug 2020 plemented by a non-Clifford gate [35], that one might measurements to perform code deformations [44, 47, 54{ implement by magic state distillation [36, 37], these oper- 58]. We show that we can encode code deformations ations are enough to recover universal quantum computa- within a resource state such that after we measure most tion [38, 39]. Owing to its high threshold error rates [38{ of its qubits, the remaining unmeasured qubits of the re- 40] and simplicity in its design [20, 41{43], the topological source state lie in the state of the deformed stabilizer cluster state is the prototypical model for robust fault- code, thereby completing a fault-tolerant operation. We tolerant measurement-based quantum computation [44]. also show how we can compose many of these operations, The number of two-qubit entangling operations we re- that we will often call channels, to realize longer compu- quire per physical qubit to realize our gate set are no tations and more complex code deformations. more demanding than those to realize the standard topo- Since the work of Raussendorf et al. it has been logical cluster state. We therefore expect that the Clif- shown [59] how to map a special class of stabilizer ford gates we propose will maintain the high threshold codes, known as Calderbank-Shor Steane(CSS) stabilizer of the topological cluster state. As such, our model is codes [60, 61], onto a measurement-based model. We particularly appealing for experimental realization. have extended beyond this work by showing how to per- 2 form computations via code deformations on arbitrary precisely we have stabilizer codes from the measurement-based perspec- tive. Further, in Ref. [62] it was proposed that lattice Pj = 11 11 11 P 11 11; (1) surgery [63, 64] can be mapped onto a measurement- | ⊗ ⊗{z · · · ⊗ } ⊗ ⊗ | ⊗ ·{z · · ⊗ } j−1 n−j based computational model. As we will discuss, the spe- cific model proposed in Ref. [62], together with any other where 11 is the two-by-two identity matrix and P = model of code deformation [38, 39, 44], can be repro- X; Y; Z is a Pauli matrix. duced using our framework. We finally remark on recent The gauge group describes a code space specified by work [65, 66] on measurement-based quantum computa- its stabilizer group tion where the topological cluster-state model is general- ized to find robust codes, and Refs. [35, 67] where a new ( ) ; (2) schemes for universal fault-tolerant measurement-based S/C G \G quantum computation are proposed based using three- where ( ) denotes the centralizer of a group within dimensional codes [68, 69]. C G G n which consists of all elements of n that commute The remainder of this article is organised as follows. withP all elements of . With the stabilizerP group defined We begin by reviewing notation we use to describe quan- we specify the codeG space as the subspace spanned by a tum error-correcting codes in Sec. II. After introducing basis of state vectors where some basic notation and the concept of foliation, we sum- j i marize the results of our paper and give a guide to the s = (+1) ; (3) reader to parse the different aspects of our model in j i j i Sec. III. Then, in Sec. IV develop a microscopic model for the one-dimensional cluster state as a simple instance of for all stabilizers s . By definition, the stabilizer group must satisfy 21 S . foliation, and we consider parity measurements between − 62 S separate foliated qubits. In Sec. V we use the microscopic We also consider a generating set of logical operators framework we build to show how a channel system can = ( ) . The group is generated by the logical L C G nG L propagate an input stabilizer code unchanged. We explic- Pauli operators Xj, Zj with 1 j k, such that Xj ≤ ≤ itly demonstrate this by foliating the twisted surface code anti-commutes with Zl if and only if j = l. Otherwise all model in Sec. VI. In Sec. VII we go on to show that we can generators of the group of logical Pauli operators com- manipulate input states with a careful choice of channel mute with one another. systems. We demonstrate this by showing we can per- The logical operators generate rotations within the form Clifford gates and prepare noisy magic states with code space of the stabilizer code. We will frequently make the foliated surface code before offering some concluding use of the fact that logical operators L; L0 such that remarks. Appendices describe extensions of our results L0 = sL with s have an equivalent action2 L on the code and proofs of technical details. Appendix A describes space. This follows2 S from the definitions given above. We an alternative type of foliated qubit, Appendix B gives thus use the symbol ` ' to denote that two operators proofs of technical theorems we state in the main text, are equivalent up to multiplication· ∼ · by a stabilizer oper- Appendix C discusses foliated subsystem codes and Ap- ator. For instance, with the given example we can write pendix D gives a generalization of the foliated stabilizer L0 L. ∼ codes we propose. It is finally worth noting that the special Abelian sub- class of subsystem codes, namely stabilizer codes [50, 71], are such that = up to phases. S G II. QUANTUM ERROR CORRECTION A. Transformations and compositions of codes Here we introduce the notion of a subsystem code [70] that we use to describe the foliated systems of interest. A It will be important to make unitary maps between subsystem code is a generalization of a stabilizer code [50, subsystem codes. Given the generating set of two differ- 71] where not all of the logical operators of the code are ent stabilizer codes and with elements r and used to encode logical information. The logical operators s , the stabilizerR group S = is generated2 R by for each of the disregarded logical qubits are known as elements2 S r 1; 1 s .T We alsoR ⊗ use S the shorthand gauge operators.