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Quantum Pin Codes Christophe Vuillot, Nikolas P Quantum Pin Codes Christophe Vuillot, Nikolas P. Breuckmann To cite this version: Christophe Vuillot, Nikolas P. Breuckmann. Quantum Pin Codes. 2019. hal-02351417 HAL Id: hal-02351417 https://hal.archives-ouvertes.fr/hal-02351417 Preprint submitted on 6 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Quantum Pin Codes Christophe Vuillot, Nikolas P. Breuckmann To cite this version: Christophe Vuillot, Nikolas P. Breuckmann. Quantum Pin Codes. 2019. hal-02351417 HAL Id: hal-02351417 https://hal.archives-ouvertes.fr/hal-02351417 Submitted on 6 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. 1 Quantum Pin Codes Christophe Vuillot and Nikolas P. Breuckmann Abstract—We introduce quantum pin codes: a class of quan- a transversal gate, but not any code admits such gates. More tum CSS codes. Quantum pin codes are a vast generalization generally, for many codes in the CSS code family it is possi- of quantum color codes and Reed-Muller codes. A lot of the ble to fault-tolerantly implement Clifford operations, which structure and properties of color codes carries over to pin codes. Pin codes have gauge operators, an unfolding procedure and their are all unitary operations preserving Pauli operators under stabilizers form multi-orthogonal spaces. This last feature makes conjugation. Clifford operations by themselves do not form them interesting for devising magic-state distillation protocols. a universal gate set. Several techniques to obtain a universal We study examples of these codes and their properties. gate set, by supplementing the non-Clifford T gate to Cliffords for example, have been devised [18], [19], among which magic state distillation is currently the most promising candidate. I. INTRODUCTION In this work we introduce a new class of CSS codes, which HE REALIZATION of a fault-tolerant universal quantum we call quantum pin codes. These codes form a large family T computer is a tremendous challenge. At each level of while at the same time have structured stabilizer generators, the architecture, from the hardware implementation up to the namely they form multi-orthogonal spaces. This structure is quantum software, there are difficult problems that need to be necessary for codes to admit transversal phase gates and it can overcome. Hovering in the middle of the stack, quantum error be leveraged to obtain codes that can be used within magic correcting codes influence both hardware design and software state distillation protocols. Moreover the construction of pin compilation. They play a major role not only in mitigating codes differs substantially from previous approaches making noise and faulty operations but also in devising protocols it an interesting space to explore further. to distill the necessary resources granting universality to an In section II, after introducing some notations and termi- error corrected quantum computer [1]. The study and design nology, we define quantum pin codes, explain their relation of quantum error correcting codes is therefore one of the to quantum color codes and give some concrete approaches major tasks to be undertaken on the way to universal quantum to construct them. In section III, we discuss the conditions computation. for transversal implementation of phase gates on a CSS code A well-studied class of quantum error correcting codes are and magic state distillation. In section IV, we investigate the Calderbank-Shor-Steane codes (CSS codes) [2], [3], which are properties of pin codes. Finally in section V, we study concrete a kind of stabilizer quantum codes [4], [5]. The advantage of examples of pin codes obtained from Coxeter groups and chain CSS codes is their close connection to linear codes which complexes as well as applications for magic state distillation. have been studied in classical coding theory. A CSS code can be constructed by combining two binary linear codes. Roughly speaking, one code performs parity checks in the Pauli X-basis II. PIN CODES and the other performs parity checks in the Pauli Z-basis. Not any two binary linear codes can be used: it is necessary that A. Terminology and formalism any two pairs of code words from each code space have to have even overlap. Common classical linear code constructions, e.g. Consider D+1 finite, disjoint sets, (C0;:::;CD) which we random constructions, do not sit well with this restriction and call levels. The elements in each of the levels are called pins. can therefore not be applied to construct CSS codes. Several If a pin c is contained in a set Cj then j is called the rank arXiv:1906.11394v2 [quant-ph] 7 Aug 2019 families of CSS codes have been devised based on geometrical, of c. Since all the Cj are disjoint each pin has a unique rank. homological or algebraic constructions [6]–[17], however, it is Consider a (D + 1)-ary relation on the D + 1 levels still open which parameters can be achieved. C0;:::;CD, that is to say a subset of their Cartesian product Besides being able to protect quantum information, quantum F ⊂ C0 ×· · ·×CD. The tuples in the relation F will be called error correcting codes must also allow for some mechanism to flags. process the encoded information without lifting the protection. A subset of the ranks, t ⊂ f0;:::;Dg, is called a type. We It is always possible to find some operations realizing a desired will consider tuples of pins coming from a subset of the levels action on the encoded information but these operations may selected by a type t and call them collection of pins of type t. spread errors in the system. One should restrict themselves A collection of pins of type t = fj1; : : : ; jkg, is therefore an to fault-tolerant operations which do not spread errors. For element s 2 Cj1 ×· · ·×Cjk . Note that we can interchangeably instance, acting separately on each qubit of a code cannot view a collection of pins as a tuple or a set as long as no two spread single qubit errors to multi-qubit errors. This is called pins come from the same level in the set. C. Vuillot, QuTech, TU Delft, [email protected]. This paper was pre- We now define specific subsets of flags, called pinned sets, sented at QEC19 in London. using projections. N. P. Breuckmann, University College London, [email protected]. 2 Definition 1 (Projection of type t). Given a set of flags F and Proposition 2 (Pinned set decomposition). Let s be a type t 0 0 a type t = fj1; : : : ; jkg, the projection, Πt, is defined as the collection of pins and let t be a type containing t, i.e. t ⊃ t. natural Cartesian product projection acting on the flags, F The pinned set Pt(s) is partitioned into some number, say m, of pinned sets, each characterized by a type t0 collections of Πt : F ! Cj1 × · · · × Cjk 0 pins containing s, i.e. sj ⊃ s, (c0; : : : ; cD) 7! (cj ; : : : ; cj ): 1 k m G 0 Note that the projection of empty type, Π;, is also well Pt(s) = Pt0 (sj): defined: for any f 2 F we have Π;(f) = (). j=1 Definition 2 (Pinned set). Let F be a set of flags, s be Proof. Let s be a type t collection of pins and let t0 be a type 0 a collection of pins of type t and Πt be the corresponding such that t ⊃ t. Define the following set of collections of pins projection as defined above. We define the pinned set of type t S = Πt0 (Pt(s)) ; and collection of pins s, Pt(s), as the preimage of s under the projection Πt, then it is the case that −1 Pt(s) = Πt (s) ⊂ F: G 0 Pt(s) = Pt0 (s ): In words: a pinned set is the set of flags whose projection of s02S a given type yields a given collection of pins, . A definition t s Indeed, since t0 ⊃ t and 8s0 2 S; s0 ⊃ s we have that 8s0 2 of a pinned set which is equivalent to the one given above is 0 S; Pt0 (s ) ⊂ Pt(s) showing the right to left inclusion. The left to right inclusion follows from the definition of S. Finally the Pt(cj1 ; : : : ; cjk ) = F \C0 · · ·×f cj1 g×· · ·×f cjk g×· · · CD: (1) fact that the union is disjoint follows from Prop. 1. The pinned set with respect to the empty type is none other than the full set of flags, F . For convenience, we will refer B. Definition of a (x; z)-pin code to a pinned set defined by a collection with k pins as a Equipped with the notions layed out in the previous section, k-pinned set. we now construct quantum codes. They are defined by a choice If one wants to form a mental image one can imagine a of flags F and two natural integers x and z which fulfill the pin-board with pins of different colors for each levels on it.
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