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Quantum Topological Error Correction Codes Are Capable of Improving the Performance of Clifford Gates Daryus Chandra, Zunaira Babar, Hung Viet Nguyen, Dimitrios Alanis, Panagiotis Botsinis, Soon Xin Ng, and Lajos Hanzo

Abstract—The employment of quantum error correction codes under certain conditions we can transform various classes of (QECCs) within quantum computers potentially offers a reli- powerful classical error correction codes into their quantum ability improvement for both quantum computation and com- counterparts, such as Quantum Turbo Codes (QTCs) [8], [9], munications tasks. However, incorporating quantum gates for performing error correction potentially introduces more sources Quantum Low-Density Parity-Check (QLDPC) codes [10], of quantum decoherence into the quantum computers. In this [11], and Quantum Polar Codes (QPCs) [12], [13]. However, scenario, the primary challenge is to find the sufficient condition several challenges remain, hindering the immediate employ- required by each of the quantum gates for beneficially employing ment of these powerful QSCs in quantum computers. Firstly, QECCs in order to yield reliability improvements given that the the reliability of the state-of-the-art quantum gates is still quantum gates utilized by the QECCs also introduce quantum decoherence. In this treatise, we approach this problem by significantly lower compared to classical gates. For exam- firstly presenting the general framework of protecting quantum ple, the reliability of a two-qubit quantum gate is between gates by the amalgamation of the transversal configuration of 90.00% 99.90% across various technology platforms, such quantum gates and quantum stabilizer codes (QSCs), which can as spin− electronics, photonics, superconducting, trapped-ion, be viewed as syndrome-based QECCs. Secondly, we provide and silicon [14]–[22]. Similar to the classical domain, in- examples of the advocated framework by invoking quantum topological error correction codes (QTECCs) for protecting both voking an error correction code within a quantum computer transversal Hadamard gates and CNOT gates. The simulation requires an additional building block. However, adding an and analytical results explicitly show that by utilizing QTECCs, additional block for error correction also implies that we the fidelity of the quantum gates can be beneficially improved, unavoidably introduce an additional source of decoherence provided that quantum gates satisfying a certain minimum into the quantum computers, since the encoder and decoder depolarization fidelity threshold (Fth) are available. For instance, for protecting transversal Hadamard gates, the minimum fidelity of QSCs are also composed of quantum gates. Secondly, the values required for each of the gates in order to attain fidelity powerful QSCs such as QTCs, QPCs, and QLDPC codes improvements are 99.74%, 99.73%, 99.87%, and 99.86%, when require a very long codewords in order to operate close to they are protected by colour, rotated-surface, surface, and toric the quantum hashing bound. In other words, they require a codes, respectively. These specific Fth values are obtained for very high number of physical quantum bits (qubits) in order a very large number of physical qubits (n → ∞), when the to correct numerous errors. Additionally, the qubits have a quantum coding rate of the QTECCs approaches zero (rQ → 0). Ultimately, the framework advocated can be beneficially exploited relatively short coherence time [23], and therefore, the error for employing QSCs to protect large-scale quantum computers. correction procedure has to be completed before the ensemble Index Terms—quantum error correction codes, quantum sta- of the qubits starts decohering. Hence, utilizing QSCs having bilizer codes, fault-tolerant, quantum topological codes, quantum a high number of qubits for correcting many errors has the gates potential threat of encountering more erroneous qubits before the error correction procedure is even completed. Thirdly, the I.INTRODUCTION state-of-the-art architecture of quantum computers imposes an additional challenge, where the interactions among the qubits Quantum error correction codes (QECCs) have the potential are ideally limited to the nearest neighbour qubits, which can of offering a promising reliability improvement in quantum be arranged by introducing a lattice-based topological archi- computers. A popular member of the QECC family is consti- tecture. The aforementioned challenges impose limitations on tuted by quantum stabilizer codes (QSCs) [1]–[7], which can creating a fault-tolerant error correction architecture. be viewed as the class of syndrome-based QECCs. The field The quest for creating fault-tolerant gates was initialized of QECCs benefitted from a rapid pace development because when the notion of transversal configuration was introduced The authors are with the School of Electronics and Computer Sci- for quantum gates [24], [25]. The concept of transversal gates ence, University of Southampton, Southampton, SO17 1BJ, UK (email: relies on a parallel set of identical quantum gates imposed {dc2n14, zb2g10, hvn08r, da4g11, pb1y14, sxn, lh}@ecs.soton.ac.uk). The financial support of the EPSRC under the grant EP/L018659/1, that of for carrying out the operation of a single quantum gate. the European Research Council, Advanced Fellow Grant QuantCom and that The fact that transversal quantum Clifford gates preserve the of the Royal Society’s Global Research Challenges Fund (GRCF) is gratefully stabilizer formalism after the conjugation operation creates an acknowledged. Additionally, the authors acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the opportunity for employing a wide range of QSCs for protecting University of Southampton, in the completion of this work. transversal quantum gates. However, the challenges we have 2 described earlier suggest that the family of quantum topo- despite relying only on a single stabilizer measurement for logical error correction codes (QTECCs) is the most suitable each of the stabilizer operators?” Arguably, the answer is yes, candidate for protecting the transversal quantum gates. The provided that certain conditions are satisfied. In this treatise, motivation behind combining the QTECCs with the transver- we answer this question by quantifying both the limit of sal configuration of quantum gates is that they conveniently the quantum depolarizing probability that can be tolerated by complement each other. More explicitly, the transversal imple- two-dimensional QTECCs and the minimum fidelity threshold mentation has the benefit of stabilizer preservation, while the required by each of the quantum gates, so that the two- QTECCs provide localized stabilizer measurements as detailed dimensional QTECCs investigated can be beneficially invoked later in Section II, which consequently provides the benefit for improving the reliability of the transversal quantum gates. of a constant number of qubit interactions as we increase Against the aforementioned background, our contributions the number of physical qubits. Thus, the benefits provided are: by topologically inspired stabilizer formalism will not be 1) We present a general framework of transversal quantum affected by the transversal implementation of quantum gates. gates protected by QSCs for gate-based quantum com- Hence, the localized action of stabilizer operators amongst the puters by performing only a single stabilizer measure- adjacent qubits, which offers fault-tolerance, is still preserved ment for each of the stabilizer operators for achieving even after the desired quantum operation has been carried out the error correction. Furthermore, we also provide a by the transversal quantum gates. tutorial-style portrayal of this in the context of both However, the amalgamation of the QSCs and the transversal the transversal Hadamard and controlled-NOT (CNOT) quantum gates can only be implemented for quantum Clifford gates protected by a simple quantum repetition code. gates, but not for the universal set of quantum gates. In 2) We highlight the implementation of two-dimensional order to expedite the universality of quantum computation, QTECCs conceived for protecting transversal Hadamard a set of fault-tolerant non-Clifford quantum gates has to be as well as CNOT gates and demonstrate the qubit error conceived. Fortunately, this can be achieved also using QSCs ratio (QBER) improvements attained compared to the by invoking the so-called magic state distillation [26], which unprotected quantum gates. is however, beyond the scope of this paper. Nonetheless, the 3) We present the upper-bound and lower-bound qubit protection of Clifford gates is of significant importance, since error rate (QBER) performance of transversal quantum the development of large scale quantum computers heavily Clifford gates protected by QSCs. relies on QSCs. This in turn hinges on the protection of 4) Based on the analytical QBER performance, we found quantum Clifford gates in the face of quantum decoherence. the probability threshold and the fidelity threshold to The threshold theorem of [27] was introduced for demon- be satisfied by the quantum gates in order to obtain strating that a quantum computation task at a vanishingly the benefit of reliability improvement upon employing low QBER can be carried out even with the presence of QTECCs. These thresholds mark the ultimate limit for errorneous quantum gates, with the aid of QECCs as long as the physical realization of fault-tolerant quantum gates the error rate imposed by the quantum gates is below a certain relying on QTECCs. probability value. Since the errorneous quantum gates may The rest of this treatise is organized as follows. We provide also result in errorneous stabilizer measurements, typically a brief review of QTECC in Section II and then we proceed a repeated stabilizer measurements are usually required for with the formulation of our framework in Section III. This is performing the error correction. The number of stabilizer followed by the design examples of QSC-protected Hadamard measurements required grows, as we increase the number of and CNOT gates in Section IV, where we invoke a simple physial qubits utilized for QECCs. This specific problem mo- quantum repetition code as our QSC. In order to evaluate tivated the emergence of the so-called single-shot QSCs [28]– the performance of our proposed framework, in Section V, [31]. Indeed, upon assuming that the syndrome values acquired we present the decoherence model utilized in our simulations. from the syndrome mesurements are not reliable, we can still Then in Section VI, we quantify the performance of QTECC- achieve a vanishingly low QBER for a quantum computation protected transversal Hadamard gates and CNOT gates both or communication task, provided that we only perform a single in terms of their QBER and fidelity along with the derivation stabilizer masurement for each stabilizer operator. However, of the QBER upper-bound and the lower-bound depolarization in order to conceive single-shot QSCs, the code constructions fidelity threshold. Finally, we conclude in Section VII. should satisfy certain criteria. One of them is related to the growth of the minimum distance as a function of the number of physical qubits. However, unfortunately, the quantum coding II.QUANTUM TOPOLOGICAL ERROR CORRECTION CODES rate of the family of two-dimensional QTECCs [32]–[35] tends (QTECCS) to zero for long codeword [36]–[38]. We also have to mention In order to make this treatise to be self-contained, this in this context that the idea of extracting reliable syndrome section provides a brief review of QSCs, more specifically values from potentially error-infested stabilizer measurements QTECCs, in terms of their encoding process, stabilizer formal- is directly related to the study of the single-shot QSCs ism, and also stabilizer measurements. In this treatise, we only as detailed in [39]–[43]. Hence, one can ask the judicious present the necessary description required for understanding question: “Can we still use the two-dimensional QTECCs how the QTECCs operate. Motivated readers might like to of [32]–[35] for achieving fault-tolerant quantum computation refer to [38], [44]–[46] for more detailed information. 3

A. A Brief Review of Quantum Information definition: In the quantum domain, the fundamental unit of information 0 + 1 0 1 + = | i | i, = | i − | i, (8) is represented by the quantum bit (qubit). In contrast to the | i √2 |−i √2 classical domain, where each of the classical bits can only and vice versa. The computational basis can also be expressed carry the value of or , a qubit can be described as a linear 0 1 as an equal-weight superposition of the vectors from the combination of and , or in other words, it can be described 0 1 Hadamard basis, which is formulated as by their superposition. However, this superposition state will + + + collapse to the corresponding classical state of 0 or 1 upon 0 = | i |−i, 1 = | i − |−i. (9) observation or measurement. More specifically, the quantum | i √2 | i √2 state of a single qubit ψ can be formally expressed as | i In order to extend the concept of quantum information to ψ = α0 0 + α1 1 , α0, α1 C, (1) | i | i | i ∈ multi-qubit systems, we have to describe the Kronecker tensor where the probability of obtaining the classical state 0 and 1 product, which is also often referred to as the tensor product. 2 2 upon measurement is given by α0 and α1 , respectively. Explicitly, for a pair of matrices P and Q having (a b) ele- | | | | × Since the values of α0 and α1 are associated with probability ments and (x y) elements, respectively, the resultant tensor 2 2 × values, the unitary constraint of α0 + α1 = 1 is applied. product is a matrix having (ax by) elements formulated as A single-qubit system can also| be| represented| | as a two- × p11Q p Q p1bQ dimensional Hilbert space, where the computational basis ··· 1(b−1) vectors 0 and 1 are defined as follows: p21Q p2(b−1)Q p2bQ | i | i  . ···. . .  P Q = ...... 1 0 ⊗   0 = , 1 = . (2) p Q p Q p Q | i 0 | i 1  (a−1)1 ··· (a−1)(b−1) (a−1)b       pa1Q pa(b−1)Q pabQ  Therefore, the quantum state of a single qubit given in Eq. (1),  ···   (10) can also be represented as For instance, a two-qubit system is represented by the tensor product between a pair of two-element vectors given in Eq. (3). ψ = α0 0 + α1 1 | i | i | i More explicitly, let us consider two qubits having the state 1 0 = α + α of ψ1 = α0 0 + α1 1 and ψ2 = β0 0 + β1 1 . The 0 0 1 1 | i | i | i | i | i | i     superimposed state then can be described as follows: α0 = , α0, α1 C. (3) α0β0 α1 ∈   α0 β0 α0β1 ψ = ψ1 ψ2 = =   Representing the basis vector of ‘0’ using the notation 0 | i | i ⊗ | i α1 ⊗ β1 α1β0 | i     and the basis vector ‘1’ using the notation 1 is referred to as α1β1 the ket notation. The terminology ket comes| i from the bra-ket   α0β0 00 + α0β1 01 + α1β0 10 + α1β111 , (11) notation [47], where the bra notation refers to the ψ notation, ≡ | i | i | i | i h | while ket notation is used for ψ . The relationship between where α0, α1, β0, β1 C. It can be observed that a two-qubit ∈ ψ and ψ is defined as follows:| i state is a superposition of all four possible states that can be | i h | † generated by a pair of classical bits i.e. 00, 01, 10 and 11. ψ = ψ , (4) 2 2 h | | i Additionally, the unitary constraint of α0β0 + α0β1 + 2 2 | | | | where the notation † indicates the conjugate transpose of α1β0 + α1β1 = 1 still holds. The tensor product of ψ | | | | ψ . Explicitly, based| i on the vector representation of Eq. (3) a pair of two-element vectors yields a vector consisting of 2 N and| i the definition of Eq. (4), we have 2 elements. Hence, the N-qubit system produces all of 2 possible states that can be generated by an N-bit sequence. ψ = α∗ α∗ , (5) h | 0 1 If i is the decimal representation of an N-bit sequence, the where α∗ denotes the complex conjugate
of α. Therefore, the N-qubit superposition state can be expressed by the Dirac following equality holds: notation as follows: 2N −1 2N −1 ψ ψ ψ ψ = 1. (6) 2 ψ = αi i where αi C and αi = 1. (12) h | i ≡ h | · | i | i | i ∈ | | i=0 i=0 In general, a pair of vectors can be used as the basis vectors, X X as long as both of them are orthonormal, normalized and As an instance of a very special case, where we have N qubits also mutually orthogonal. Apart from the above computational all having the + state provides us with all the equal-weight basis, the following Hadamard basis is also widely used in the superposition of| 2iN possible states generated by all possible field of QECCs: combination of N-bit sequences as follows: ⊗N 1 1 1 1 + + 1 + 2 + N + = , = . (7) | i ≡ | i ⊗ | i ⊗ · · · ⊗ | i | i √2 1 |−i √2 1 2N −1    −  1 The Hadamard basis can be viewed as the equal-weight super- = i , (13) √ N | i position of the computational basis according to the following 2 i=0 X 4 where the superscript N of + represents the N-fold as the simultaneous bit and phase-flip are equiprobable [9], tensor product. It can also⊗ be observed| i that the probability of [11], [44] and hence, we have pX = pZ = pY = p/3. Con- obtaining each of the 2N quantum states upon measurement sequently, the probability of having no error on an individual 1 I in the computational basis is equal to 2N . This particular state qubit is p = (1 p). This particular model is referred to as is often used as the initial quantum state for various quantum the symmetric quantum− depolarizing channel. computing algorithms, such as Shor’s quantum factoring algo- rithm [48], [49] and Grover’s quantum search algorithm [50], C. Quantum Encoder [51] as well as for quantum error correction. Similar to the classical domain, quantum error correction B. Quantum Decoherence can be achieved by incorporating redundancy into the encoded word in order to maintain the level of reliability of quantum Quantum computers are composed by numerous quantum computation. In quantum domain, due to the constraints of no- gates and also their interconnections, which are not immune cloning theorem as well as the collapsing of quantum states to environmental impairments. Consequently, due to the dele- into classical states, the syndrome-based QECCs are widely terious effects of quantum decoherence, the resultant quantum acknowledged for mitigating the errors imposed by quantum state at the output may not be the desired outcome of the decoherence without actually measuring the quantum informa- quantum computation. The quantum decoherence inflicting a tion within the qubits. This specific syndrome-based QECCs single qubit error can be represented by the Pauli group 1, P are also referred to as quantum stabilizer codes (QSCs). In which defines the discrete set of possible unitary transforma- order to carry out the error correction task, the QSC schemes tions imposed on a single qubit. Explicitly, the Pauli group 1 P require the k logical qubits to be encoded to n physical qubits is defined as with the aid of (n k) auxiliary qubits. This specific task is − 1 = eP : P I, X, Y, Z , e 1, i , (14) carried out by the so-called quantum encoder . The quantum P { ∈ { } ∈ {± ± }} encoder maps the k logical qubit based stateVψ and (n k) which is closed under multiplication. The unitary matrices X auxiliaryV qubits to a unique codespace of| thei n physical− and Z represent the bit-flip and the phase-flip, respectively, qubit based state ψ , which is defined asC follows: while the matrix Y represents the simultaneous bit-flip and | i ⊗(n−k) I = ψ = ( ψ 0 ) . (19) phase-flip. Finally, the identity unitary matrix denotes the C {| i V | i ⊗ | i } absence of error. These Pauli matrices are defined as follows: Therefore, the action of quantum encoder is reminiscent of 1 0 0 1 the generator matrix G of classical errorV correction codes, I = , X = , 0 1 1 0 which have a direct relationship with the associated parity     0 i 1 0 check matrix (PCM) H used for predicting the number and the Y = , Z = . (15) i −0 0 1 position of errors. Similarly, in quantum domain, the function    −  of predicting the number and the position of quantum errors Since the qubits whose quantum states only differ in their is carried out by a set of stabilizer operators Si . global phase can be deemed to be equivalent, the reduced ∈ S ∗ Throughout this treatise, we consider gate-based quantum Pauli group 1 denoted by is often used for the sake of P P1 computation [46], [52], noting that there are many other simulating the quantum errors by exploiting the Pauli-to-binary models for quantum computation, such as measurement-based isomorphism [46], which is defined as quantum computation [53], quantum cellular automata [54], ∗ topological quantum computation [55], and adiabatic quantum 1 = I, X, Y, Z . (16) P { } computation [56]. Therefore, in this treatise, we refer the For the quantum state of an N-qubit system, the quantum readers to the method presented in [57]–[59] for creating an decoherence effects may be described by the Pauli group n, efficient quantum encoder for QSCs. To elaborate a little which is represented by an n-fold tensor product of Pas V 1 further, given the so-called stabilizer operators Si , we defined below: P ∈ S can transform the stabilizer operators Si into their clas- sical binary PCM using the so-called Pauli-to-binary∈ S isomor- n = P1 P2 Pn Pj 1 , (17) P { ⊗ · · · ⊗ | ∈ P } phism [44], [60]. The resultant binary PCM H derived from where the index j represents the j-th qubit of a system the stabilizer formalism Si is utilized for constructing the ∈ S having n physical qubits. An operator P n transforms quantum encoder of the associated QECC . Since we deal ∈ P the legitimate quantum state ψ into an erroneous quantum with the family ofV gate-based quantum computers,C naturally, | i state ψ , as formally described below: the quantum encoder is also composed of quantum gates. | i The method specifiedV in [57]–[59] is applicable to ψ = P ψ . (18) b | i | i both Calderbank-Shor-Steane (CSS) [3]–[5] and non-CSS The quantum decoherence may inflict an individual bit-flip codes [6], [7]. However, for colour codes [34], which belong to b (X), a phase-flip (Z), as well as a simultaneous bit-flip and the QTECCs family, they can also be classified further as the phase-flip (Y) error with a probability of pX, pZ, and pY, member of a more specific category of quantum CSS codes, respectively. These deleterious effects may be inflicted upon namely, the dual-containing CSS codes. For dual-containing each of the qubits within the N-qubit block. Most of the CSS codes, a specific technique can be invoked for creating literature on QSCs assume that the bit-flip, phase-flip as well the associated quantum encoder . This method of generating V 5 the quantum encoder of dual-containing CSS codes has been 1 2 3 detailed in [10], [61].V It is important to note that upon using the method detailed in [10], [57]–[59], [61], the number of 1 2 quantum gates required to construct a quantum encoder for a QSC is linearly proportional to the number of physicalV 14 2 5 3 qubits [57],C [58]. However, we can actually dispense with the quantum en- coder by simply preparing all the auxiliary qubits in the 6X 7 8 state ofV + and then perform stabilizer measurements before 3 4 the transversal| i gates as suggested in [62], [63]. By initializing all the (n k) auxiliarry qubits to the + states, we create the 4 95 10 6 equal-weight− superposition of all possible| i combination from the (n k)-bit sequences. Therefore, in order to obtain the Z valid encoded− state of the physical qubits, we only require the stabilizer measurements for projecting the initialized space into a specific code space based on the classical bits obtained 115 12 6 13 from the stabilizer measurements. This will be further elabo- rated further on Section IV. Fig. 1: An example of physical qubits arrangement on a rectangular lattice structure. D. Stabilizer Formalism A QSC can be viewed as the a syndrome-based QECC [1], on the fact that the vertices and the plaquettes (faces) of a [44]. The stabilizer operator Si is an n-tupple Pauli ∈ S simple graph may be used to define their stabilizer generators. operator in n preserving the legitimate encoded state of the P Let us refer to the rectangular lattice structure seen in Fig. 1. physical qubits as encapsulated below: The physical qubits are represented by the black filled-circles

Si ψ = ψ . (20) located on the edges of the lattice, while the red filled-squares | i | i laying on the vertices of the graph define the X stabilizer For the erroneous physical qubits state given in Eq. (18), the generators, and finally the blue filled-triangles located on the action of stabilizer operators Si upon the state ψ can ∈ S | i plaquettes constitute the Z stabilizer generators. Formally, the be formulated as follows: stabilizer generators Gi of the QTECCs are defined as b follows: ∈ S ψ ,SiP = PSi Si ψ = | i (21) | i ( ψ ,SiP = PSi. Av = Xi , Bp = Zi. (24) −|b i − The eigenvaluesb of 1 play the similar roles to the syndrome i∈vertexY (v) i∈plaquetteY (p) b bits 1, 0 of the classical± error correction codes. Based on More specifically, the QTECC based on the rectangular lattice { } these syndrome values, an error recovery operator Ri n structure given in Fig. 1 has the stabilizer generators as may be applied to the erroneous physical qubits. Hence,∈ Pthe portrayed in Table I1. One observations that we can make is QSC carries out its error correction procedure by predicting that given the stabilizer generators given in Table I, all loops both the number and the position of erroneous qubits without on the lattice structure are indeed stabilizer operators i , S ∈ S actually measuring and altering the legitimate quantum state. since multiplying two or more stabilizers generators Gi ∈ S Due to the definition given in Eq. (20), stabilizer operators provide us with legitimate stabilizer operators Si . ∈ S Si naturally inherit the commutative property as follows: For a rectangular lattice exemplified in Fig. 1, we have 13 ∈ S physical qubits, six X stabilizer operators, and six Z stabilizer SiSj ψ = SjSi ψ = ψ , Si,j , i = j. (22) | i | i | i ∀ ∈ S 6 operators. The number of logical qubits k encoded into the It also implies that the product between the stabilizer operators physical qubits n can be calculated as follows: Si,Sj yields another legitimate stabilizer operator as ∈ S k = n , (25) described below: − |G| where is the cardinality of the stabilizer generators of the Si ψ = Sj ψ = SiSj ψ = ψ , Si,j , i = j, (23) |G| | i | i | i | i ∀ ∈ S 6 stabilizer group . Consequently, we have a single logical suggesting that the group of stabilizer operators is closed qubit encoded usingS the specific arrangement seen in Fig. 1. S under multiplication. The quantum coding rate rQ of a QSC [n, k, d] is defined as C Let us denote the QSC maping k logical qubits into n the ratio between the number logical qubits k to the number physical qubits having a minimum distance d as [n, k, d]. n−k C Hence, a QSC [n, k, d] will have 2 stabilizer operators, 1 C In Table I, the shortened representation of stabilizer operators is used for where all the stabilizer operators Si can be generated by simplifying the original representation of stabilizer operators. For example, ∈ S the shortened version of S = X X X X is used for simplifying S = only (n k) stabilizer generators Gi by the multiplication 3 4 6 7 9 3 − ∈ S I1 ⊗ I2 ⊗ I3 ⊗ X4 ⊗ I5 ⊗ X6 ⊗ X7 ⊗ I8 ⊗ X9 ⊗ I10 ⊗ I11 ⊗ I12 ⊗ I13. For between the stabilizer generators Gi. A popular family of the rest of the paper, we always use the original representation for stabilizer QSCs is constituted by the QTECCs, whose constructions rely operators, unless it is stated otherwise. 6

TABLE I: The stabilizer generators (Gi) for QSCs based on with all the stabilizer operators Si , but not an element of rectangular lattice structure depicted in Fig. 1. This specific ∈ S , the weight of Pi determines the minimum distance of the construction maps a single logical qubit into 13 physical QSCS defined by the rectangular lattice of Fig. 1. Therefore, qubits. The code has a minimum distance of 3 (d = 3), which we conclude that based on the stabilizer generators given in means that it is capable of correcting a single qubit error. Table I, the minimum distance of the QSC defined by the rectangular lattice of Fig. 1 is d = 3. Furthermore, the size Gi Av Gi Bp of the lattices defining the stabilizer operators Gi can be used for calculating the minimum distance d, the∈ number S G1 X1X2X4 G7 Z1Z4Z6 of logical qubits and physical qubits, as well as the quantum G2 X2X3X5 G8 Z2Z4Z5Z7 coding rate rQ. More details on this topic are provided in [38]. G3 X4X6X7X9 G9 Z3Z5Z8

G4 X5X7X8X10 G10 Z6Z9Z11

G5 X9X11X12 G11 Z7Z9Z10Z12 X G6 X10X12X13 G12 Z8Z10Z13 ψ | i of physical qubits , which can be expressed as follows [5]: b n 0 H H M k | i rQ = . (26) (a) n In general, QSCs will suffer from having a lower coding rate than their classical counterparts, since QSCs have to correct not only bit-flip (X) errors, which is also the type of error Z ψ experienced in classical domain, but also the phase-flip (Z) | i errors as well as simultaneous bit-flip and phase-flip (Y) errors [38], [60]. b The error correction capability t of a QSC [n, k, d] can be 0 M C | i determined by its minimum distance d as follows: (b) d 1 t = − . (27) Fig. 2: Stabilizer measurements for QTECCs based on rectan- 2   gular lattice depicted in Fig. 1. (a) The X stabilizer measure- Therefore, in order to verify the error correction capability of ment. (b) The Z stabilizer measurement. a QSC , first we have to evaluate the minimum distance d C based on the stabilizer operators Si . Let the normalizer ∈ S ( ) n be represented by the set of operators Pi n, so N S ∈ P† ∈ P that PSiP = Sj for all Si and i is not necessarily ∈ S ∈ S equal to j. It is plausible that all the stabilizer operators Si are automatically in ( ). Now, we are interested in the∈ specificS set of operatorsN inS the normalizer ( ) that does not belong to the stabilizer operators , whichN isS denoted by ( ) . The minimum distance ofS a QSC is equal to dN, ifS and− Sonly if ( ) contains no elementsC with weight N S − S less than d, where the weight of a Pauli operator Pi n is given by the number of non-identity Pauli operators.∈ In P other words, the minimum distance of a QSC can be defined by Fig. 3: An example of qubit arrangements in colour codes. C the minimum weight of the operators Pi, which commutes The black circles on the vertices represent the physical qubits, with all stabilizer operators Si , but it is not an element while the red squares on the plaquettes constitute both the X of . ∈ S and Z stabilizer operators. InS case of the rectangular lattice structure of Fig. 1, a good example of such an operator Pi is constituted by the chains Since the stabilizer operators of QTECCs are defined based connecting the two boundaries of the lattice. To elaborate a on the underlying lattice structure, the stabilizer measurement little further, let us consider a Pauli operator P represented can be performed locally in the lattice, i. e. without invoking by the shortened version P = X2X7X12 connecting the more distant qubits in the lattice. The X and Z stabilizer boundaries of the lattice of Fig. 1. It can be readily checked measurements of the rectangular lattice structure given in that this specific Pauli operator P commutes with all the Fig. 1 are portrayed in Fig. 2. It can be clearly observed that stabilizer generators Gi , but it cannot be represented due to the constraints imposed by the lattice structure, each of ∈ S as the product of any stabilizer generators Gi . Since the physical qubits will have to interact with at most two X ∈ S Pi represents the lowest-weight Pauli operator Pi commuting stabilizer measurements and two Z stabilizer measurements. 7

The QTECCs defined over square lattice structure exemplified The transformation carried out by the Hadamard gates map in Fig. 1 are referred to as surface codes [33]. However, the the quantum state in computational basis into the Hadamard square lattice is not the only structure defining the stabilizer basis, as described below: formalism of QTECCs. Another example is colour codes [34], 0 + 1 where the stabilizer generators G are defined over trian- H 0 = | i | i + , i | i √2 ≡ | i gular and hexagonal lattice structures∈ S as depicted in Fig. 3. In 0 1 Fig. 3, the physical qubits are represented by the black circles H 1 = | i − | i (31) on the vertices of the lattice, while the stabilizer measurements | i √2 ≡ |−i are represented by the red squares on the plaquette. The and vice versa. Based on the conjugation of Eq. (29), we arrive main difference is that the plaquette of the lattice is used for at the following transformations: defining both X and Z stabilizer operators. Furthermore, we HXH† = Z, can observe that each physical qubit only interacts with at most † three stabilizer measurements and this number will remain HZH = X, constant even for a larger lattice dimension. This property HYH† = Y. (32) is quite important in this treatise, since we want to keep the − number of interacting qubits as low as possible. In this treatise, One way to interpret the transformation given in Eq. (32) is for the sake of comparison, we will use four types of QTECCs, that a bit-flip (X) error before the Hadamard gate is equivalent namely surface codes [33], colour codes [34], rotated-surface to a phase-flip (Z) error after the Hadamard gate. Similarly, a codes [35], and toric codes [32]. phase-flip (Z) error before the Hadamard gate can be treated as an X error after the Hadamard gate. Another example is the phase gate (S), which is defined by the unitary transformation of III.PROTECTING TRANSVERSAL GATES 1 0 S = . (33) 0 i In this section, we present the design of QSC-protected   quantum gates along with the pivotal theory in quantum Based on Eq. (29) and on the unitary matrix of S in Eq. (33), information processing required for formulating the proposed we arrive at the following transformations: framework. SXS† = Y SZS† = Z SYS† = X (34) A. Quantum Clifford Gates − The quantum gates manipulating the state of the qubits are Moreover, the relationship given in Eq. (29) can also be represented using the unitary transformation U, which satisfies adopted to the model of the conjugation over two-qubit quan- tum gates. An example of two-qubit quantum gate is CNOT, U †U = I. (28) whose unitary transformation is defined by the following Since the quantum gates themselves also potentially impose unitary matrix: the deleterious effect of quantum decoherence, which is rep- 1 0 0 0 resented by the Pauli group n, the evolution of quantum 0 1 0 0 P CNOT = . (35) decoherence through the quantum gates can be described using 0 0 0 1 the conjugation of the unitary transformation. 0 0 1 0   Let us assume that a unitary transformation of M is applied   to a quantum state of N ψ , where N is also a unitary A CNOT gate is a two-qubit quantum gate, where the first transformation that has been| appliedi previously. Therefore, the qubit is called control qubit and the second qubit is referred final quantum state is given by MN ψ . Since M is a unitary to as target qubit. The value of target qubit is flipped if and transformation, thus we have M †M| =i I. The quantum state only if the value of control qubit is equal to one. Based on of MN ψ can be transformed as follows: Eq. (29), the two-qubit conjugation over the CNOT gate can | i be readily formulated as follows: MN ψ = MNM †M ψ | i | i (CNOT)(X I)(CNOT)† = X X, = VM ψ , (29) ⊗ ⊗ | i (CNOT)(I X)(CNOT)† = I X, † ⊗ ⊗ where we have V = MNM , which is the conjugate of N (CNOT)(Z I)(CNOT)† = Z I, under the unitary transformation M. Consequently, Eq. (29) ⊗ † ⊗ † (CNOT)(I Z)(CNOT) = Z Z. (36) implies that the unitary transformation MNM after the ⊗ ⊗ operator M acts similarly to the unitary transformation N We can also interpret the result as an error transformation or before the operator M. For instance, a Hadamard gate is an error propagation process, illustrated in Fig. 4. It can be defined by a unitary matrix as follows: observed that a bit-flip or X-type error imposed on the control 1 1 1 qubit before the CNOT gate will propagate to the target qubit H = . (30) Z √2 1 1 after the CNOT gate. Meanwhile, a -type error inflicted upon  −  8

the target qubit before the CNOT gate will impose another Z- Si designed for stabilizing the legitimate state of physical type error on the control qubit after the CNOT gate. qubits∈ S are no longer valid after the application of the unitary transformation Uf . Fortunately, we are able to conceive an XXI I effective stabilizer formalism and effective inverse encoder for successfully circumventing the problem2. correlate commute To elaborate a little further, let us observe the basic model for QSC portrayed of Fig. 6(a) and the basic scheme of I XX X protecting quantum gates using QSC, as depicted in Fig. 6(b). We would like to highlight the main differences between the (a) (b) conventional QSC scheme of Fig. 6(a) and the QSC scheme conceived for protecting the quantum gates shown in Fig. 6(b). Z Z I Z Firstly, in Fig. 6(a), the stabilizer operators Si are designed for stabilizing the state of physical qubits ψ∈, S while commute correlate | i the effective stabilizer operators Si in Fig. 6(b) are constructed for stabilizing the state of∈ physicalS qubits after I I Z Z the unitary transformation Uf ψ =b ψ2 .b Consequently, the | i | i (c) (d) recovery procedures of Ri based on stabilizer operators is also modified according∈ to R the effective stabilizer operatorsS Fig. 4: The evolution of two-qubit unitary transformation over into the effective recovery operators R . Secondly, the the CNOT quantum gate, which is the circuit level interpre- i inverseS encoder † of Fig. 6(a) is designed∈ toR recover the orig- tation of Eq. (36). For example, an X-type error imposed on V inalb state of the logical qubit ψ , whileb in Fig.b 6(b), the block the control qubit will propagate to the target qubit after the | i † is invoked for restoring the state of the logical qubits ψ , CNOT gate, as we can observe in (a). By contrast, a Z-type 0 Vwhich has been transformed by the unitary transformation| Ui error inflicting the target qubit will propagate to the control f intob Uf ψ0 . qubit after the CNOT gate, which is shown in (d). | i Now, we proceed to specifically elaborate further on the In this treatise, we limit our discussions to quantum gates scheme proposed for protecting quantum gates using QSCs, as belonging to the Clifford group, since the stabilizer formalism shown in Fig. 6(b). Let us commence with the logical qubits in the state of ψ representing the input of the quantum circuit. In of QTECCs will be preserved under conjugation [64], [65]. | i The theory of protecting quantum circuits using QSCs has order to encode the logical qubits, we require (n k) auxiliary qubits (ancillas) all in the state of 0 . We assume− that fresh been widely investigated [24], [25], [66]–[70] and in this trea- | i tise, we provide a comparative study of the quantum circuits ancillas are always available provided by a quantum memory. protected by QSCs, specifically by the family of QTECCs. Hence, the state of the logical qubits and of the ancillas can The most reasonable way of embedding QSCs into quantum be expressed as ⊗(n−k) gates is by the transversal implementation of quantum gates. ψ0 = ψ 0 . (37) The physical interpretation of the transversal implementation | i | i ⊗ | i The quantum encoder of Fig. 6(b) maps the input state of of QTECCs, specifically that of CNOT gates, is illustrated V ψ0 into the state of encoded logical qubits or physical qubits in Fig. 5. In the physical implementation, we can imagine | i ψ1 as follows: having a pair of lattice structures arranged on a wafer, where | i ψ1 = ψ0 . (38) each of the physical qubit layers is encoded using a QTECC | i V| i scheme. The first qubit from the upper layer acts as the control The unitary operation Uf of Fig 6(b), which is the unitary qubit, while the first qubit from the lower layer serves as the transformation that we wish to protect, maps the physical target qubit. It is followed by the second, the third, and all the qubits in the state of ψ1 onto ψ2 as follows: remaining physical qubits from the upper as well as the lower | i | i layer. ψ2 = Uf ψ1 = Uf ψ0 . (39) | i | i V| i It is important to note that the unitary transformation of Uf B. Design Formulation may impose quantum errors on the encoded state ψ2 , as P | i The employment of QSCs for protecting the quantum cir- it will be shown later in Section V. cuits indeed increases the reliability of quantum computing, as Due to the unitary tranformation Uf of Fig. 6(b) after the shown in [61]. However, in order to avoid a perpetual encoding quantum encoder , the stabilizer operators of Si are no V ∈ S and decoding process throughout the quantum circuit before longer applicable to the encoded state of ψ2 , because the sta- the number of errors exceeds the error correction capability bilizer formalism was designed for stabilizing| i the encoded S of the QSC, we present a more efficient framework, where state of ψ1 . Since the aim of this scheme is to protect the state we encode the logical qubits at the input of the quantum | i circuit and decode them afterwards at the output of the 2The term effective means that the stabilizer formalism and the inverse encoder at the output of computational step takes into account both the initial computational step. However, the unitary transformation Uf stabilizer formalism and quantum encoder when we encode the logical qubits applied to the state of physical qubits alters the legitimate state at the input and the unitary transformation applied upon the input physical of physical qubits. Therefore, the initial stabilizer operators qubits. 9

control qubits

CNOT CNOT CNOT φ | 1i . . 1 2 3 . . 1 2 3 4 5

6 7 8 4 5 6 7 8 target 9 10 qubits 9 10 φ 11 12 13 | 2i . . . . 11 12 13

CNOT CNOT CNOT

control qubits φ1 | i target qubits φ2 U | i f (a) (b) Fig. 5: The physical interpretation of the transversal CNOT gates is portrayed in (a), while (b) depicts its interpretation at the circuit level. In the physical implementation of (a), we can imagine two layers of physical qubits, where on each layer the physical qubits are arranged over a lattice structure portrayed in Fig. 1. The CNOT interactions are performed accordingly on each of the pairs of physical qubits. The CNOT interactions between the physical qubits number #4 to #10 are removed from the figure for the sake of avoiding obfuscation. The physical layout portrayed in (a) can be translated into circuit level interpretation of (b), where it represents a chain of CNOT gates.

of ψ2 instead of ψ1 , a different stabilizer formalism, which QSCs to non-transversal quantum Clifford gates. Since the | i | i we refer to as the effective stabilizer operators Si , should stabilizer operators are changed, there is a chance that the ∈ S be designed for stabilizing the state of ψ2 by taking into resultant stabilizer operators Si are associated with a | i ∈ S account the unitary transformation Uf . Given theb formulationb QSC exhibiting a higher error correction capability. This is in Eq. (29), we represent the effective stabilizer operator S as reminiscent of the idea of codeb deformationb presented in [71]– follows: [74]. Defining Uf for the non-transversal configurations of the b quantum Clifford gates in order to achieve an increased error ψ2 = Uf ψ1 | i | i correction capability constitutes a substantial open research = Uf S ψ1 question on its own right. Hence, we set aside this issue | i † for our future research. Thirdly, the resultant set of stabilizer = Uf SUf Uf ψ1 | i operators Si constitutes no longer a set of n-tupple Pauli = SUf ψ1 ∈ S | i operators. This means that the unitary transformation Uf is = S ψ2 , (40) not representedb b by a Clifford quantum gate. More specifically, | i b if the unitary transformation Uf is represented, for example, where S is the effective stabilizerb for ψ2 , which is defined by a Tofolli gate [46], our QSCs cannot be directly invoked by | i † for protecting the quantum gates by the procedures presented b S = Uf SUf , (41) in this treatise. In order to tackle this problem, magic state distillation as a technique required for protecting non-Clifford for stabilizing the state of ψ2 in Fig. 6(b). b | i quantum gates was proposed in [26]. Again, in this treatise, we Given a unitary function Uf and a set of stabilizer op- focus our attention on the first possible outcome, since we are erators Si , there are three possible outcomes due to dealing with transversal quantum Clifford gates. Nevertheless, the effective∈ stabilizer S formalism of Eq. (41). Firstly, the set the aforementioned remaining implications of the effective of stabilizer operators does not change ( = ). In other stabilizer formalism are also interesting research directions. S S words, the overall stabilizer operators are preserved. This Next, after performing the stabilizer measurements of Si ∈ specific result arises because the unitary transformationb Uf is , the error recovery procedure of Fig. 6(b) acts according S R constituted by certain types of transversal quantum Clifford to the effective stabilizer measurements and we obtainb the gates, such as for instance transversal Hadamard gates or predictedb legitimate physical qubitsb state of ψ2 . The final | i CNOT gates and because Si belongs to the family task of our system given in Fig. 6(b) is to perform a unitary ∈ S of dual-containing CSS-type QSCs. Secondly, the stabilizer transformation in order to transform the finalc state of ψ2 | i operators are changed ( = ). One of the possible reasons into the state of Uf ψ0 . Similar to the stabilizer formalism S 6 S | i that we arrive at the second result is because we apply the in Fig. 6(a), the inverse encoder was also designedc S V b 10

ψ ψ ψ | i | i | ′i b ψ ψ′ = ψ | i | i L| i 0 | i 0 | i . . † (n k) 0 VP R V 0 ⊗ − | i M| i 0 | i 0 | i

S (a) The basic model of QSCs implementation over the quantum depolarizing channel.

ψ ψ ψ | 1i | 2i | 2i c

...... ψ0 . . f . . . † . ψ | i VU P R V Uf| 0i b b

S

(b) The basic model for QSC-protected quantum gates where the unitary transformation Uf introduces quantum depolarizing channel. Fig. 6: The comparison between QSC for correcting the errors imposed by quantum channel without and with unitary b P transformation f . Figure (a) depicts the standard scheme for protecting the encoded state of physical qubits ψ . In this scheme, the purposeU of applying a QSC is to recover the state of logical qubit ψ by stabilizing the state ψ| usingi the | i | i stabilizer operators Si in the presence of depolarizing channel . Figure (b) portrays the modified scheme for protecting ∈ S P the encoded state of physical qubits ψ2 , which is the encoded state after unitary transformation f . In this scheme, instead | i U of stabilizing the state of ψ1 , the stabilizer operators have to stabilize the state of ψ2 . Since the stabilizer operator Si | i | i ∈ S is only valid for the encoded state of ψ1 , we have to reformulate the stabilizer operators Si for stabilizing the encoded state | i ψ2 , which we refer to as the effective stabilizer operators S1 . | i ∈ S b b for recovering the state of ψ , which is based on the inverse encoder † is given by | 1i V stabilizer formalism . Therefore, we require the effective † † S Uf ψ0 = Uf (Uf ) (Uf ) ψ0 inverse encoder , which is designed based on the effective b| i V V | i V † stabilizer operators , as seen in Fig. 6(b), to recover the state = Uf (Uf ) ψ2 S V | i of Uf ψ0 . The formulation used for describing the effective † † b = Uf U ψ2 | i V f | i b † = ψ2 , (42) V | i † where is the effective inverse encoder for ψ2 , which is V b | i b 11

defined as: stabilizer generators Si for stabilizing the quantum state ψ2 † † † | i = Uf U . (43) in Fig. 7(a) as follows: V V f b † ⊗3 ⊗3 † Finally, as seen in Eq. (42), the effective inverse encoder S1 = H (Z1 I2 Z3) H b V ⊗ ⊗ succesfully transforms the state ψ2 into the state of Uf ψ0 . † † † | i | i = (HZH
)1 (HIH )2 (HZH
)3 b b ⊗ ⊗ = X1 I2 X3, (48) ⊗ ⊗

IV. DESIGN EXAMPLES ⊗3 ⊗3 † S2 = H (Z1 Z2 I3) H † ⊗ ⊗ † † One of the strategies we may rely on for creating high- = (HZH
)1 (HZH )2 (HIH
)3 b ⊗ ⊗ reliability quantum gates is that of invoking the unitary = X1 X2 I3, (49) transformation Uf , which is a sequence of quantum gates ⊗ ⊗ arranged in a transversal fashion. For example, let us consider which are capable of correcting a single phase-flip (Z) error. Finally, to transform the physical qubits in the state of ψ2 to a quantum gate protected by 1/3-rate quantum repetition code | i as shown in Fig. 7(a). More explicitly, based on the model the state of Uf ψ0 , we design the effective inverse encoder † | i depicted in Fig. 6(b), we describe the transversal implementa- of based on Eq. (43) and we obtain the quantum circuit V tion of both the Hadamard and of the CNOT gates protected portrayed in Fig. 7(a). Therefore, the final state of Uf ψ0 can | i by the 1/3-rate quantum repetition code. be describedb as follows: ⊗n ⊗(n−k) Uf ψ0 = H ( ψ 0 ) | i | i ⊗ | i = H⊗k ψ (H⊗(n−k) 0 ⊗(n−k)) A. Transversal Hadamard Gates | i ⊗ ⊗ | i = H⊗k ψ + ⊗(n−k). (50) | i ⊗ | i First, we would like to elaborate on the transversal imple- The last term of + ⊗(n−k) can be discarded during measure- mentation of Hadamard gates protected by a 1/3-rate quantum ment, since it is| noi longer entangled. In this example, we repetition code. Let us refer to Fig. 7(a). The procedure arrive at + + . Finally, from the design example specified begins with encoding the state of a single logical qubit in Fig. 7(a),| wei acquire the desired output in the state of ψ = α 0 + β 1 and two auxiliarry qubits in the state of | i | i | i H⊗k ψ = H ψ = α + + β , given that we have k = 1 0 into the state of physical qubits of ψ1 with the aid of | i | i | i |−i | i | i and ψ = α 0 + β 1 . quantum encoder as follows: | i | i | i V ⊗2 ψ1 = ψ 0 In this treatise, another technique of utilizing QSC for | i V | i ⊗ | i = (α 0 + β 1 ) 0 ⊗2 protecting the transversal quantum Clifford gates without V | i |
i ⊗ | i the quantum encoder is presented. In order to relax the = α 000 + β 111
assumption of havingV a perfect quantum encoder , all the | i | i V α 0 L + β 1 L, (44) (n k) auxiliarry qubits required for creating the encoded ≡ | i | i state− of the physical qubits are initialized to the + state where 0 = 000 , 1 = 111 , and the quantum encoder L L and then followed by stabilizer measurements as suggested| i for this| i mapping| i is| portrayedi | byi the part marked by in in [62], [63]. The quantum circuit of tranversal Hadamard VFig. 7(a). The encoded state of the physical qubits givenV in gates protected by a -rate quantum repetition code is Eq. (44) is stabilized by the stabilizer operators generated by 1/3 portrayed in Fig. 7(b). Let us denote the encoded state of Si: physical qubits α 000 + β 111 of Eq. (44) as ψ1 and the | i | i | i S1 = Z1 I2 Z3, encoded state of physical qubits α + ++ + β of ⊗ ⊗ Eq. (47) as . The quantum state| iof Fig.| − 7(b) −−i can S2 = Z1 Z2 I3, (45) ψ2 ψ0 ⊗ ⊗ then be expressed| i as | i which can be invoked for correcting a single bit-flip (X) error. ψ0 = (α 0 + β 1 ) + + . (51) The Hadamard gates, which carry out the unitary transforma- | i | i | i ⊗ | i ⊗ | i tion we wish to protect, are arranged transversally, where the |0i+|1i Given that + = √ , we can expand and rewrite the quan- Hadamard gate is applied to each of the encoded logical qubits | i 2 tum state ψ of Eq. (51) in terms of ψ = α 000 +β 111 as seen in Fig. 7(a). Hence, the unitary transformation U can 0 1 f as follows:| i | i | i | i be expressed as ⊗n Uf = H , (46) ψ0 = ψ1 + X2 ψ1 + X3 ψ1 + X2X3 ψ1 , (52) | i | i | i | i | i where the superscript n of H represents the n-fold tensor where the notation of X indicates that Pauli matrix X is ⊗ i product. The unitary transform Uf transforms the quantum applied to the i-th qubit. The current quantum state of ψ0 | i state ψ1 into quantum state ψ2 as follows: is in a form of superposition of ψ1 states. More specifically, | i | i | i the action of stabilizer measurements (Si ) will result ψ2 = α + ++ + β . (47) ∈ S | i | i | − −−i in collapsing the quantum state ψ0 into one of the following | i Based on Eq. (41), the stabilizer operators of Eq. (45) and the possibilities: ψ1 , X2 ψ1 , X3 ψ1 , and X2X3 ψ2 with equal | i | i | i | i unitary transformation Uf of Eq. (46), we obtain the effective probabilities. The resultant collapsed quantum state ψ1 is | i 12

f † ψ = α 0 + β 1 V U V | i | i | i H b 0 ψ | i H Uf | 0i 0 R | i H

ψ ψ ψ ψ | 0i | 1i | 2ib | 2i c S

(a) Example of transversal implementation of Hadamard gates protected by a 1/3-rate quantum repetition code with quantum encoder V.

V f † ψ = α 0 + β 1 U b V | i | i | i H b 0 H ψ | i H f 0 R R U | i 0 H | i H

ψ ψ ψ ψ | 0i | 1i | 2ib | 2i c S S

(b) Example of transversal implementation of Hadamard gates protected by a 1/3-rate quantum repetition code with repeated stabilizer measurements. b Fig. 7: The realization of the transversal implementation of quantum gates protected by the 1/3-rate quantum repetition code. The unitary operator Uf in these examples represents the transversal implementation of Hadamard gates.

determined by the 1 values gleaned from the stabilizer defined by ψ = φ1 φ2 and our desired output is ± | i | i ⊗ | i measurements Si of Fig. 7(b). Hence, the error recovery CNOT( φ1 φ2 ), where φ1 acts as the control qubit and ∈ S | i ⊗ | i | i is capable of transforming resultant collapsed quantum state φ2 serves as the target qubit. The quantum encoder of R | i V back into the legitimate quantum state ψ1 . Hence, its action Fig. 8(a) encodes each of the logical qubits in the state of φ1 | i | i is similar to that of the scheme seen in Fig. 7(a) relying on the and φ2 independently, yielding the physical qubits in the state | i quantum encoder . Next, the action of the unitary operator of ψ1 = φ1 φ2 , where φ1 and φ2 are the encoded V | i | i ⊗ | i | i | i Uf , which is represented by the transversal Hadamard gates of logical qubits of φ1 and φ2 , respectively. Therefore, the | i | i Eq. (46), transforms the quantum state ψ1 into the quantum stabilizer operators Si provided for the physical qubits in | i state ψ2 as follows: the state of ψ1 in Fig. 8(a) are generated by the following | i stabilizer generators:| i ψ2 = Uf ψ1 | i | i = α + ++ + β . (53) S1 = Z1 I2 Z3 I4 I5 I6, | i | − −−i ⊗ ⊗ ⊗ ⊗ ⊗ S2 = Z1 Z2 I3 I4 I5 I6, From this point onwards, both the effective recovery operators ⊗ ⊗ ⊗ ⊗ ⊗ as well as the effective quantum inverse encoder † of the S3 = I1 I2 I3 Z4 I5 Z6, R V ⊗ ⊗ ⊗ ⊗ ⊗ schemes operating either with or without quantum encoder , S4 = I1 I2 I3 Z4 Z5 I6. (54) V ⊗ ⊗ ⊗ ⊗ ⊗ areb identical. Therefore, the error correction performanceb of the two schemes is also expected to be identical.

In this example, the quantum gate that we try to protect is B. Transversal CNOT Gates the CNOT gate. Therefore, the unitary transformation Uf of For our next example, we aim for conceiving the transversal Fig. 8 represents the unitary matrix of the transversal CNOT implementation of CNOT gates protected by the 1/3-rate gates. Now, let us denote the unitary matrix of the n transversal n quantum repetition code. The model is depicted in Fig. 8. CNOT gates as CNOT . Therefore, the unitary transformation For the scheme portrayed in Fig. 8(a), the initial state is Uf of Fig. 8, can be expressed as a 64 64-element matrix × 13

f † φ = α 0 + β 1 | 1i 1| i 1| i V U V

0 b | i

0 | i

f ψ0 φ = α 0 + β 1 R | 2i 2| i 2| i U | i 0 | i

0 | i b

ψ ψ ψ ψ | 0i | 1i | 2i | 2i c S (a) Example of transversal implementation of CNOT gates protected by a 1/3-rate quantum repetition code with quantum encoder V.

V f † φ = α 0 + β 1 U b V | 1i 1| i 1| i b 0 H | i

0 H | i ψ φ = α 0 + β 1 f 0 | 2i 2| i 2| i R R U | i

0 H | i

0 H b | i

ψ ψ ψ ψ | 0i | 1i | 2i | 2i c S S (b) Example of transversal implementation of CNOT gates protected by a 1/3-rate quantum repetition code with repeated stabilizer measurements. b Fig. 8: The realization of the transversal implementation of CNOT gates protected by the 1/3-rate quantum repetition code. The unitary operator Uf in these examples represents the transversal implementation of CNOT gates. The QSC-protected quantum gates can be realized with quantum encoder or with repeated stabilizer measurements. Naturally, the scheme without quantum encoder is favourable since it potentiallyV eliminates the error propagation imposed by the CNOT gates. V by expanding the following transformation: the following stabilizer generators: 3 U = CNOT S1 = Z1 I2 Z3 I4 I5 I6, f ⊗ ⊗ ⊗ ⊗ ⊗ = 000 000 I I I + 001 001 I I X S2 = Z1 Z2 I3 I4 I5 I6, | ih | ⊗ ⊗ ⊗ | ih | ⊗ ⊗ ⊗ b ⊗ ⊗ ⊗ ⊗ ⊗ + 010 010 I X I + 011 011 I X X S3 = Z1 I2 Z3 Z4 I5 Z6, | ih | ⊗ ⊗ ⊗ | ih | ⊗ ⊗ ⊗ b ⊗ ⊗ ⊗ ⊗ ⊗ + 100 100 X I I + 101 101 X I X | ih | ⊗ ⊗ ⊗ | ih | ⊗ ⊗ ⊗ S4 = Z1 Z2 I3 Z4 Z5 I6. (56) + 110 110 X X I + 111 111 X X X. b ⊗ ⊗ ⊗ ⊗ ⊗ | ih | ⊗ ⊗ ⊗ | ih | ⊗ ⊗ ⊗ (55) b Based on the commutativity property of stabilizer opera- Based on the stabilizer generators Si in Eq. (54) and tors, where the multiplication among stabilizer operators will the unitary transformation Uf of Eq. (55), we can obtain produce another valid stabilizer operator, we may rewrite the the effective stabilizer operators Si by applying Eq. (41), list of stabilizer generators given in Eq. (56). Hence, by which formulated for protecting the state of ψ2 in Fig. 8(a). multiplying S and S also by multiplying S and S , we | i 1 3 2 4 Explicitly, the effective stabilizer operatorsb Si is generated by obtain the following effective stabilizer operators Si generated b b b b b b 14 by the following stabilizer generators: as follows:

S1 = Z1 I2 Z3 I4 I5 I6, ψ0 = ψ1 + X2 ψ1 + X3 ψ1 + X2X3 ψ1 ⊗ ⊗ ⊗ ⊗ ⊗ | i | i | i | i | i S2 = Z1 Z2 I3 I4 I5 I6, + X5 ψ1 + X2X5 ψ1 + X3X5 ψ1 + X2X3X5 ψ1 b ⊗ ⊗ ⊗ ⊗ ⊗ | i | i | i | i S3 = I1 I2 I3 Z4 I5 Z6, + X6 ψ1 + X2X6 ψ1 + X3X6 ψ1 + X2X3X6 ψ1 b ⊗ ⊗ ⊗ ⊗ ⊗ | i | i | i | i S4 = I1 I2 I3 Z4 Z5 I6. (57) + X5X6 ψ1 + X2X5X6 ψ1 + X3X5X6 ψ1 b ⊗ ⊗ ⊗ ⊗ ⊗ | i | i | i + X2X3X5X6 ψ1 , (61) b | i where the notation of Xi indicates that Pauli matrix X is applied to the i-th qubit. Notice that the stabilizer operators of Eq. (54) are associated with 16 possible classical syndrome As we can observe, the resultant effective stabilizer op- vectors, where each of the syndrome vectors is associated with erators Si of Eq. (57) are identical with the stabilizer ∈ S one of the superimposed states given in Eq. (61). When the operators Si of Eq. (54). Consequently, it demonstrates stabilizer measurements are performed, the superposition state the convenience∈ S of exploiting the transversal implementation b b of Eq. (61) will result in collapsing the quantum state ψ0 into of CNOT gates. Since the effective stabilizer operators of | i Si one of the following 16 possibilities: ψ1 , X2 ψ1 , X3 ψ1 , Eq. (57) are identical to the stabilizer operators of Eq. (54), | i | i | i Si X2X3 ψ1 , X5 ψ1 , X2X5 ψ1 , X3X5 ψ1 , X2X3X5 ψ1 , the quantum circuit of the effective inverse encoder † of | i | i | i | i | i b X6 ψ1 , X2X6 ψ1 , X3X6 ψ1 , X2X3X6 ψ1 , X5X6 ψ1 , Fig. 8 is also identical to the inverse encoder † designedV | i | i | i | i | i V X2X5X6 ψ1 , X3X5X6 ψ1 , X2X3X5X6 ψ1 , where the for transforming the state ψ1 to the state ψ0 . Finally,b | i | i | i | i | i collapsed quantum state is determined by the classical syn- the desired state for the output physical qubits based on the drome vector obtained from stabilizer measurements. There- transversal configuration given in Fig. 8 can be formulated as fore, we can apply the recovery operator to obtain the n R ⊗(n−k) ⊗(n−k) encoded state of physical qubits of ψ1 of Eq. (59). After Uf ψ0 = CNOT ( φ1 0 , ( φ2 0 )) | i | i ⊗ | i | i ⊗ | i the recovery operator is carried out, the| unitaryi transformation k ⊗(n−k) = CNOT ( φ1 , φ2 ) 00 , (58) is applied to the quantum state , which transforms the | i | i ⊗ | i Uf ψ1 quantum state of ψ into quantum| statei ψ as follows: where the last term represents the auxiliary qubits in the state | 1i | 1i of 00 ⊗(n−k) that can be discarded after measurement, since ψ2 = Uf ψ1 . (62) they| arei no longer entangled with the logical qubits in the | i | i k Similar to transversal Hadamard gates, from this point on- state of CNOT ( φ1 , φ2 ). Given that we have k = 1 and n = 3 for our example| i | ini Fig. 8, we succesfully protect the wards, all the effective stabilizer measurements, error recovery operators, and inverse encoder of both Fig. 8(a) and Fig. 8(b) unitary transformation of CNOT( φ1 , φ2 ) at the end of our scheme. Therefore, we have outlined| i the| i general framework are identical. Hence, we expect the error correction perfor- of QSC-protected transversal quantum gates. mance of the two schemes illustrated in Fig. 8 to be identical.

V. ERROR MODEL In order to show the benefit of utilizing the framework ex- emplified in Section IV, we have to opt for a realistic quantum Additionally, we also present the scheme of protecting decoherence model of the system in order to produce the most transversal CNOT gates without the quantum encoder in realistic performance results. However, several assumptions Fig. 8(b). Instead of using the quantum encoder portrayedV V have to be made in order to justify the proposed error model. in Fig. 8(a), we can also utilize the scheme of Fig. 8(b), where Based on [75]–[78], we stipulate the idealized simplifying the quantum encoder is replaced with stabilizer masurements V assumption that quantum decoherence may be imposed by to create the legitimate encoded quantum state of the physical single-qubit quantum gates, two-qubit quantum gates, as well qubits. More specifically, the quantum state of the physical as by the deleterious effects of the stabilizer measurements. qubits ψ1 of Fig. 8(a) is given by | i In order to consider the decoherence inflicted by multiple components within our framework, we offer the following two ψ1 = (α1 000 + β1 111 ) (α2 000 + β2 111 ) | i | i i| i ⊗ | i i| i propositions: = α1α2 000 000 + α1β2 000 111 | i| i | i| i Proposition 1. An error-infested quantum encoder and + α2β1 111 000 + β1β2 111 111 . (59) V | i| i | i| i a unitary transformation Uf can be modeled as a perfect In Fig. 8(b), the physical qubits are prepared in the quantum quantum encoder and a perfect unitary transformation Uf followed by the quantumV channel P . To elaborate a little state of ψ0 as follows: n | i further, the encoder of QSCs can be fully∈ P constructed from the ψ0 = (α1 0 + β1 1 ) + + quantum gates belonging to the Clifford group. Consequently, | i | i | i ⊗ | i ⊗ | i (α2 0 + β2 1 ) + + . (60) we can use the formulation of Eq. (29), which is illustrated in ⊗ | i | i ⊗ | i ⊗ | i Fig. 4, in order to model the error propagation throughout the |0i+|1i Given that + = √ , we can expand and rewrite the quantum encoder. More explicitly, let us provide an example | i 2 quantum state ψ0 of Eq. (60) in terms of ψ1 of Eq. (59) using the quantum encoder of Steane’s 7-qubit code, which | i | i 15

X X X X Z Z Z Z

X X

X X

Z H H Z

Z H H Z

H H

(a) The error propagation of a single bit-flip (X) error at the (b) The error propagation of a single phase-flip (Z) error at the input of first qubit. input of first qubit. Fig. 9: The quantum encoder for Steane’s 7-qubit code can be implemented purely consisting of quantum gates from the Clifford group. Therefore, a faulty quantum encoder of any QSC can be assumed to be modelled as a perfect quantum encoder followed by the quantum depolarizing channel. A single bit-flip (X) error at the input can be transformed into three bit-flip errors at the output of quantum encoder as shown in (a). Similarly, a phase-flip (Z) inflicted at the input of quantum encoder V may propagate resulting three phase-flipV error at the output as depicted in (b). is equivalent to the quantum encoder of the distance-3 colour A. Source of Decoherence code, as portrayed in Fig. 9. We can observe from Fig. 9 that By considering the potential sources of quantum- a single bit-flip (X) error at the input of the quantum encoder decoherence based on our proposed framework and also the is transformed into three separated bit-flip (X) errors, which nature of error propagation, we conclude that diverse sources are accumulated at the output of the encoder . Similarly, a of decoherence can be efficiently modelled as the accumulated phase-flip (Z) at the input of the quantum encoderV propagating quantum decoherence before and after the transversal quantum across the quantum encoder is transformed into three different gates U , as explicitly shown in Fig. 11. To elaborate a phase-flip errors at three different locations at the output of the f little further, the quantum decoherence before the transversal quantum encoder . Hence, in general, the severity of error quantum gates is constituted by P that corrupts propagation withinV the quantum encoder is determined by 1 n the physical qubits in the state of ψ ∈by P X, Z and Y- the number of two-qubit quantum CliffordV gates composing it. 1 type errors independently with depolarizing| i probability of Proposition 2. An error-infested stabilizer measurement of pX = pY = pZ = p /3. Similarly, the quantum decoherence the QSC protecting the transversal quantum gates of Fig. 6(b) 1 after the transversal gates is denoted by P2 n), which is can be substituted by perfect stabilizer measurement and the ∈ P described by the depolarizing probability of p2, corrupting the quantum channel P after the transversal configuration of physical qubits in the state of ψ2 . quantum gates, which∈ isP illustrated in Fig. 10. This is also the | i It is important to note that each of the error operators P and natural consequences of the conjugation of Eq. (29). Since 1 P may encapsulate several sources of quantum decoherence. the X stabilizer operators anti-commute with the Z Pauli 2 For example, observe from Fig. 11 that the error operator P operator, only Z-type errors are considered for X stabilizer 1 represents the decoherence from the quantum encoder of measurements. Equivalently, since the Z stabilizer operators V Fig. 6(b), which is denoted by Pv. The error operator of Pv anti-commute with the X Pauli operator, only X-type errors ∈ n is characterized by the depolarizing probability of pv. The affect the result of Z stabilizer measurements. P error operator of P2 n encapsulates both the quantum ∈ P decoherence imposed by the quantum gates Pu as well as the Z Z quantum decoherence imposed by the stabilizer measurement Pm. If the error operator Pu is specified by the depolarizing Z Mx Mx probability of pu and the error operator Pm is specified by the depolarizing probability of pm, the error operator P2 can be Mz Mz determined by Pm Pu = P2 n. · ∈ P (a) Error on X stabilizer measurement. Let us now consider the various error models available in X X the literature. Most of the error models used for evaluating the performance of QSCs designed for protecting quantum gates Mx Mx have three parameters, namely the error probability imposed

X Mz Mz by single-qubit quantum gates denoted by pa, the error proba- (b) Error on Z stabilizer measurement. bility imposed by two-qubit quantum gates denoted by pb, and the error probability imposed by the stabilizer measurements Fig. 10: An erroneous stabilizer measurement can be equiva- denoted by pm. The standard error model of [76] assumes that lently replaced by errors before and after the quantum circuit. pa = pb = pm = p. By contrast, the balanced error model 16 of [76] assumes that the error rate imposed by the single-qubit photonics, superconducting, trapped-ion, and silicon imple- quantum gates is pa = 4p/5, as well as that by the two-qubit mentations [14]–[22]. Fortunately, based on the threshold quantum gates is pb = p, and finally that by the stabilizer theorem [27], it is possible to construct a reliable quantum measurements is given by pm = 8p/15. From an experimental computer from unreliable quantum gates, given that the gate point of view, let us consider the ion trap error model of [76], error probability is below a certain threshold value and that a where the parameters are defined by p1 = p/1000, p2 = p, sufficiently high overhead is allowed. Therefore, most of the and pm = p/100. Since in this treatise we model the quantum studies on the QECCs aim for finding the specific gate error decoherence imposed on each of the physical qubits as an probability pa, pb, and pm so that the QECCs do become individual and independent binary symmetric channel [9], [10], capable of significantly improving the reliability of quantum we have the model parameters of pa = pm = p and pb 2p. computers. A more detailed description of our error model used in≈ our investigations will be provided in the following subsection. The various error models characterize different technol- B. Faulty Quantum Gates ogy platforms available for developing quantum computers. In this treatise, however, we focus our attention on perfor- In this treatise, we rely on the model illustrated in Fig. 12, mance of transversal quantum Clifford gates protected by two- where each qubit experiences an independent quantum depo- dimensional QTECCs using classical simulations. Intuitively, larizing channel characterized by its depolarizing probability. we want the value of depolarizing probability to be as low The metric used for evaluating the performance of our system as possible. However, state-of-the-art quantum gates have is the qubit error rate (QBER) and the fidelity (F ). The QBER is defined as the ratio between the number erroneous qubits to relatively low fidelity ranging between 90.00% 99.90%, for various technology platforms, such as spin electronics,− the total number of qubits. For a single-qubit quantum gate, which is exemplified by the Hadamard gate H, the QBER value can be represented as

ψ ψ QBERHad = p, (63) | 1i | 2i where p is the depolarizing probability value of the single- H qubit quantum decoherence caused by the imperfection of the quantum gate. H

Pv H Pu Pm ......

H H Pg

H ψ ψ P1 f P2 | i | i U (a) The model of faulty Hadamard gate. (a) Transversal Hadamard gates. b ψ ψ | 1i | 2i

Pg ......

Pv Pu Pm ψ ψ | i | i (b) The model of faulty CNOT gate...... b Fig. 12: The unprotected faulty Hadamard and CNOT gates used as the benchmark. The Pauli channel P g is inflicted ∈ P P P after the quantum gates. 1 Uf 2 (b) Transversal CNOT gates. The reliability of a quantum gate can also be quantified Fig. 11: The potential quantum decoherence model imposed using its fidelity. Explicitly, the fidelity may be used to in our proposed system. There are several potential sources reflect the closeness of an ensemble of quantum states in the of quantum decoherences including the quantum encoder, mixed-state to the desired pure state, which is formulated as quantum gates, and stabilizer measurement, denoted by Pv, follows [79]–[81]: Pu and Pm, respectively. F = ψ ρ ψ , (64) h | | i 17 where ψ is the desired pure state and ρ is the density matrix For example, let us assume that the first depolarizing | i encapsulating the statistical characteristics of the mixed-states. channel is constituted by the error operator P1 n, which ∈ P The density matrix ρ is defined by is characterized by the depolarizing probability p1 = p, while N the second depolarizing channel is constituted by the error operator P , which is characterized by the depolarizing ρ = pi ψi ψi , (65) 2 n | ih | ∈ P i=1 probability p2 = p. Then, the probability of obtaining the Pauli X matrix I at the output of two consecutive Pauli channels can where ψi represents all the possible quantum states in the | i be expressed as follows: ensemble and pi is the probability of obtaining the quantum I I I X X Y Y Z Z state ψi , which is consequently constrained by the unity pf = p1p2 + p1 p2 + p1 p2 + p1 p2 | i N requirement of p = 1. For a quantum Clifford gate, p p p p p p i=1 i = (1 p)(1 p) + + + the relationship between fidelity and the QBER is as simple − − 3 3 3 3 3 3 as P 4             = 1 2p + p2. (72) − 3 F = 1 QBER, (66) − Consequently, we obtain the following probability of experi- where QBER is the qubit error ratio of the unprotected encing the Pauli error X, Y, and Z: gate. Consequently, we obtain the analytical expression of the X Y Z 2 4 2 initial fidelity (F ) for a Hadamard gate as a function of the pf = pf = pf = p p . (73) in 3 − 9 depolarizing probability p as follows: In other words, the effective depolarizing probability can be Fin = 1 p. (67) expressed as − 4 2 Similar to the Hadamard gate, the fidelity of the CNOT gate pf = 2p p can be defined using Eq. (66). Since a CNOT gate is a two- − 3 2p. (74) qubit quantum gate, the fidelity can be explicitly formulated ≈ as the probability of having the Pauli operator I on both the In can be concluded that a pair of depolarizing channels control qubit and the target qubit, which can be formulated as each characterized by the depolarizing probability p can be effectively viewed as a single quantum depolarizing channel Fin = (1 p)(1 p) − − associated with the aggregated depolarizing probability of = 1 2p + p2. (68) − pf = 2p. Furthermore, the effective depolarizing probability Therefore, the QBER of a CNOT gate can be rewritten as pf for c consecutive depolarizing channels, where each of follows: the channels is characterized by an identical depolarizing probability p, can be approximated as QBER = 2p p2 CNOT − 2p, (69) pf cp, (75) ≈ ≈ for p 1.The expression given in Eq. (75) is our basis for for p 1. Therefore, given that pa is the error rate or the  deriving the analytical upper-bound expressions in Section VI. QBER of a single-qubit quantum gate and pb is the error rate or the QBER of a two-qubit quantum gate, we obtain We have to emphasize once again that this is one of the the relationship between pa and pb obtained for classical desirable properties of QTECCs compared to the rest of QSC simulation utilizing BSC as follows: family, namely that the number of stabilizer measurements

pb 2pa. (70) applied to each physical qubit remains constant upon increas- ≈ ing the number of physical qubits n due to the convenient Additionally, the relationship between the error rate of a construction of the underlying lattice structure. As we can single-qubit quantum gate pa and the error rate of a stabilizer observe in Eq. (75), the effective depolarizing probability pf is measurement pm can be simply formulated as governed by the value of c and it is essentially the contribution from a number of stabilizer measurements experienced by each pm = pa. (71) of the physical qubits. In case of the QTECCs, the value of c remains constant upon increasing the minimum distance d C. Effective Depolarizing Channel of the code and the number of the physical qubits n. For instance, the number of stabilizer measurements for colour As we have described in Subsection V-A, in the QSC- codes is defined by the number of adjacent plaquettes for a protected quantum transversal gates, there are multiple sources given vertice. More specifically, each of the physical qubits of decoherence. For example, in Fig. 11, the decoherence in colour codes relies on at most six stabilizer measurements effects imposed by the transversal quantum gates and stabilizer regardless of the specific minimum distance d of the code and measurements can be modelled by multiple subsequent quan- of the number of physical qubits n. Similar findings are valid tum depolarizing channels. In order to simplify the analytical for surface codes. The number of stabilizer measurements calculations and their approximation, we introduce the notion relied upon by each of the physical qubits is defined by the of effective depolarizing channel. number of vertices and plaquettes, given a particular edge of a 18 lattice. Each of the physical qubits in surface codes relies on at most four stabilizer measurements, regardless of the minimum 100 distance d of the code and of the number of physical qubits n. 10−1

VI.SIMULATION RESULTS AND PERFORMANCE ANALYSIS

In order to evaluate the performance of the system con- 10−2 sidered, we exploited the classical-to-quantum Pauli isomor- QBER phism [9], [60]. To elaborate further, we generated two inde- Unprotected Hadamard gate (simulation) Unprotected Hadamard gate (analytical) pendent binary symmetric channels, where one of the channels 10−3 Quantum repetition code (rQ = 1/3) (simulation) modeled the bit-flip channel, while the other emulated the Quantum repetition code (rQ = 1/3) (analytical) Quantum repetition code (rQ = 1/5) (simulation) phase-flip channel. Since it is impossible to mimic identically Quantum repetition code (rQ = 1/5) (analytical) Quantum repetition code (rQ = 1/7) (simulation) Quantum repetition code (rQ = 1/7) (analytical) the actual quantum depolarizing channel using two indepen- 10−4 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 dent binary symmetric channels (BSCs), for approximating the Depolarizing probability (p) quantum depolarizing channel having an equal probability of X, Z, and Y-type of errors (pX = pZ = pY = p/3), it is Fig. 13: QBER performance of the transversal Hadamard gates widely accepted that the flip probability of each BSCs in the protected by 1/3, 1/5, and 1/7-rate quantum repetition codes. classical simulation is adjusted to pX = pZ = 2p/3 [10], [11]. We applied the depolarizing channel after the transversal con- The maximum-likelihood hard-decision decoding technique figuration of Hadamard gates, where the depolarizing channel was invoked, which was translated into a look-up table (LUT) is characterized by the depolarizing probability p. based decoder. For the full exposure of how we conceived the LUT decoder, we refer the reader to [60]. We take the value of frame error rate (FER) from the classical simulation to potray 100 the QBER, since we concern with unitary transformation on the arbitrary and unknown quantum state.

10−1

A. Simple Examples

In this section, we present the simplest scenario of transver- 10−2 sal Hadamard gates and CNOT gates protected by quantum QBER repetition codes. Let us revisit Fig. 7 and observe the associ- Unprotected CNOT gate (simulation) Unprotected CNOT gate (analytical) ated QBER performance. Based on the error model described 10−3 Quantum repetition code (rQ = 1/3) (simulation) in Section V, the quantum depolarizing channel Pg n Quantum repetition code (rQ = 1/3) (analytical) ∈ P Quantum repetition code (rQ = 1/5) (simulation) associated with the depolarizing probability p corrupted the Quantum repetition code (rQ = 1/5) (analytical) Quantum repetition code (rQ = 1/7) (simulation) Quantum repetition code (rQ = 1/7) (analytical) quantum state of the physical qubits ψ . For the simplest − 2 10 4 scenario here, we assumed that the quantum| i encoder and 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Depolarizing probability (p) the syndrome measurement were also error-free. V For the transversal Hadamard gates protected by a 1/3- Fig. 14: QBER performance of transversal CNOT gates pro- rate quantum repetition code, which is illustrated in Fig. 7, tected by 1/3, 1/5, and 1/7-rate quantum repetition codes. the simulation results are portrayed in Fig. 13. We also have We applied the depolarizing channel after the transversal included the performance results for transversal Hadamard configuration of CNOT gates, where the depolarizing channel gates protected by 1/5-rate and 1/7-rate quantum repetition is characterized by the depolarizing probability p. codes. Naturally, the quantum repetition codes can only protect the transversal Hadamard gates from one type of error, either the phase-flip or the bit-flip errors. In other words, they are We have also invoked our QSC-protected scheme for not capable of protecting the quantum gates from a more transversal two-qubit Clifford quantum gates, as exemplified realistic quantum depolarizing channel. Furthermore, reducing by the CNOT gate. We portray the performance of an un- the quantum coding rate - for quantum repetition codes this protected CNOT gate along with that of its QSC-protected only allows increasing the error correction capability for counterpart utilizing a 1/3-rate quantum repetition code, as either bit-flip or phase-flip errors - imposes increased quantum illustrated in Fig. 8, in Fig. 14. Furthermore, we also have decoherence on our system, which is shown explicitly by included the QBER performance curves of 1/5-rate and 1/7- the increase of QBER in Fig. 13. To elaborate further, upon rate quantum repetition codes. As expected, similar to transver- increasing the number of qubits, we only increase the error sal Hadamard gates, instead of being improved, the QBER correction capability for either phase-flip or bit-flip errors. performance of transversal CNOT gates is degraded upon Consequently, the deleterious effect of other types of errors reducing the quantum coding rate. Again, invoking QSCs for imposed by the quantum depolarizing channel is accumulated protecting quantum gates remains futile if we only consider and left uncorrected. protecting one type of errors. In conclusion, we have to employ 19

QSCs, which are capable of correcting the bit-flip, phase-flip −2 as well as the simultaneous bit-flip and phase-flip errors. 10

B. QTECC-Protected Transversal Hadamard Gates 10−3 In this section, we utilized the more practical QTECCs as the subtitutes for the above-mentioned quantum repetition − 10 4

codes, since the QTECCs are capable of mitigating both bit- QBER flip and phase-flip errors. More specifically, we considered colour codes [34], rotated-surface codes [35], and surface − 10 5 Unprotected Hadamard gate (simulation) codes [33], exhibiting a minimum distance of d = 3 in our Unprotected Hadamard gate (analytical) Colour code (rQ = 1/7) (simulation) model of Fig. 6(b). For this scenario, we used the assumption Colour code (upper bound) Colour code (lower bound) that the encoded physical qubits can be created fault-tolerantly, − 10 6 hence the quantum encoder was assumed to be error-free, 0 0.5 1 1.5 2 ×10−3 or we can utilize the schemeV without the quantum encoder Depolarizing probability (p) as we have described earlier in Section IV. The quantum (a) Colour code. depolarizingV channel was inflicted by the quantum unitary 10−2 operation Uf and also by the stabilizer measurements. The error operator Pu n imposed by the unitary operation ∈ P Uf was characterized by the depolarizing probability pu = p. − 10 3 However, for the stabilizer measurements, as we have men- tioned in Section II, each of the physical qubits will have a constant number of measurement for each X and Z stabilizer 10−4 operators. More explicitly, for surface codes and rotated- QBER surface codes, each of the physical qubits will have at most two interactions for each X and Z stabilizer measurements, while 10−5 Unprotected Hadamard gate (simulation) for colour codes each of the physical qubits will have at most Unprotected Hadamard gate (analytical) Rotated-surface code (rQ = 1/9) (simulation) three interactions for each X and Z stabilizer measurements. Rotated-surface code (upper bound) Rotated-surface code (lower bound) To elaborate a litlle further, due to the stabilizer measure- − 10 6 ments, we encountered four additional consecutive depolariz- 0 0.5 1 1.5 2 Depolarizing probability (p) ×10−3 ing channels after the error operator Pu imposed by the unitary operation Uf for surface and rotated-surface codes. Therefore, (b) Rotated-surface code. we had five consecutive depolarizing channels, where each 10−2 of the error operators Pi n for i = 1, 2, 3, 4, 5 was ∈ P { } characterized by the depolarizing probability pi = p. Simi- larly, for colour codes we inflicted six additional consecutive 10−3 depolarizing channels after the error operator Pu imposed by the unitary operation Uf , hence we imposed seven consecutive depolarizing channels after unitary operation Uf , where each − 10 4 of the error operators Pi n for i = 1, 2, 3, 4, 5, 6, 7 was ∈ P { } QBER also characterized by depolarizing probability pi = p.

Formally, the effective error operator Pf after c consecutive − 10 5 Unprotected Hadamard gate (simulation) error operators Pi can be expressed as Unprotected Hadamard gate (analytical) Surface code (rQ = 1/13) (simulation) c Surface code (upper bound) Surface code (lower bound) Pf = Pi for Pi n. (76) −6 ∈ P 10 i=1 0 0.5 1 1.5 2 Y Depolarizing probability (p) ×10−3 For c consecutive depolarizing channels, the j-th qubit from n physical qubits will experience independently the effective (c) Surface code. error operators Pf , which is defined as Fig. 15: QBER performance for transversal Hadamard gates c protected by distance-3 QTECCs over quantum depolariz- ∗ Pf,j = Pi, for Pi . (77) ing channel. The quantum decoherence is inflicted after the ∀ ∈ P1 i=1 transversal configuration of Hadamard gates, where the error Y c operators of Pi n are defined by pi = p. The number Since there are c consecutive depolarizing channels, the prob- i=1 ∈ P ability of obtaining identity matrix I is equal to the sum of of consecutive error operators c = 7 for colour codes and c = 5 for surfaceQ and rotated-surface codes. all probabilities from all the possible combinations of Pf,j resulting in Pauli matrix I, where we assume that pI = (1 p) and pX = pY = pZ = p/3. Therefore, the effective probability− 20

 p 2  p 3  p 4  p 5 pI = (1 − p)5 + 30(1 − p)3 + 60(1 − p)2 + 105(1 − p) + 60 f 3 3 3 3 40 160 320 256 = 1 − 5p + p2 − p3 + p4 − p5. (74) 3 9 27 81 I pf = 1 − pf 40 160 320 256 = 5p − p2 + p3 − p4 + p5. (75) 3 9 27 81

of obtaining Pauli matrix I for five consecutive depolarizing Ai values for colour codes up to minimum distance of d = channels can be expressed as shown in Eq. (74). 7. However, for non-degenerate QSCs, the analytical QBER To elaborate a little further, we can infer from Eq. (74) that performance can be approximated using the following upper- 5 (1) there are 30 possible combinations from i=1 Pi which give bound as described in [38], [60] denoted by QBERupper as us Pf,j = I consisting of two non-identity Pauli operators follows: ∗ Q Pi 1 , 60 combinations consisting of three non-identity t=b d−1 c ∈ P 2 n Pauli operators, 105 combinations consisting of four non- QBER(1) (n, d, p) = 1 pi(1 p)n−i identity Pauli operators, and finally, 60 combinations consist- upper − i − i=0   ing of five non-identity Pauli operators. Therefore, the effective n X n i n−i depolarizing probability pf for c = 5 can be formulated as = p (1 p) , (81) i − shown in Eq. (75). However, in case of p 1, the effective i=t+1    X depolarizing probability can be approximated as where for QSCs-protected transversal single-qubit gates p =

pf 5p, (76) pf , which in our case is the effective depolarizing probability ≈ defined by Eq. (75) for rotated-surface and surface codes and as we have suggested earlier in Eq. (75). by Eq. (78) for colour codes. Similarly, for c = 7, the sum of the probabilties from all of To elaborate a little further, the upper-bound of Eq. (81) the possible combinations Pf,j resulting in the Pauli matrix I characterizes the worst-case performance assuming that any can be expressed as shown in Eq. (77). Therefore, the effective syndrome associated with an error pattern beyond the error depolarizing probability pf for c = 7 can be expressed as correction capability of the quantum code is ignored, because portrayed in Eq. (78). Again, for p 1, the value of pf in  attempting to correct these errors may potentially introduce Eq. (78) can be approximated as additional errors. However, this decoding method does not exploit the full benefit of all the syndromes. For instance, for pf 7p, (79) ≈ a distance-3 surface code, we have n = 13 and k = 1. Hence, as we have suggested in Eq. (75). we have = n k = 12. Since the surface codes belong to |G| − We have simulated the system using the quantum-to- the family of CSS-type QSCs, the syndrome operator Gi classical Pauli isomorphism for simulating the QSCs. The is invoked for correcting the bit-flip and phase-flip errors∈ G simulation results are portrayed in Fig. 15. The performance separately. Consequently, for a distance-3 surface code, we of the proposed scheme is quantified using the QBER versus have six stabilizer operators dedicated to the correction of bit- the depolarizing probability p. For a minimum distance of flip errors ( z = 6) and six stabilizer operators dedicated to |G | d = 3, based on the code parameters given in Table II, the correction of phase-flip errors ( x = 6). Since each of the |G | the quantum coding rate rQ of colour codes, of rotated- syndrome measurement values is associated with an element of surface codes, as well as of surface codes are 1/7, 1/9, and the syndrome vector s, we have a total of 64 possible syndrome 1/13, respectively. In Fig. 15, we can observe the QBER vectors s. From Eq. (27), we know that a distance-3 surface performance improvement upon applying the scheme given code is capable of correcting a single qubit error caused by in Fig. 6(b) compared to the unprotected quantum gates, a both bit-flip, a phase flip, or both. Consequently, from the −3 especially for depolarizing probabilitiy values of p < 2 10 . total of 64 possible syndrome vectors s, only 14 syndrome × In general, the analytical QBER performance of QSC- vectors are actually associated with recoverable error patterns. protected single-qubit quantum gates can be calculated as More explicitly, the all-zero syndrome vector is associated follows: with an error-free quantum state of the physical qubits and n 13 syndrome vectors are associated with single bit-flip or (1) n i n−i QBERapprox(n, d, p) = 1 Ai p (1 p) , (80) phase-flip error patterns. The remaining syndromes are − i − 50 i=0 X   capable of detecting error patterns exhibiting an error weight where Ai is a real number coefficient portraying the success beyond the error correction capability of the distance-3 surface probability of correcting the error patterns having weight i. code, although the decoder cannot make a definitive error For a large number of n, finding the value of Ai for each of recovery decision from these remaining syndromes. This is i becomes an intractable problem. One example of works of because each of them may be associated with multiple error finding these Ai values specifically for colour codes can be patterns exhibiting an the identical error weight. Therefore, seen in [82], where the authors successfully characterized the in the upper-bound formulation of Eq. (81), these remaining 21

 p 2  p 3  p 4  p 5  p 6  p 7 pI = (1 − p)7 + 63(1 − p)5 + 210(1 − p)4 + 735(1 − p)3 + 1260(1 − p)2 + 966(1 − p) + 756 f 3 3 3 3 3 3 560 2270 1792 5488 3711 = 1 − 7p + 28p2 − p3 + p4 − p5 + p6 − p7. (77) 9 27 27 243 729 I pf = 1 − pf 560 2270 1792 5488 3711 = 7p − 28p2 + p3 − p4 + p5 − p6 + p7. (78) 9 27 27 243 729

TABLE II: The code parameters for various QTECCs based on the minimum distance d of the code [38].

Codes type Dimension Number of physical qubits (n) Number of stabilizers (|S|) Number of logical qubits (k)

∗ 1 2
1 2
Colour [34] d 4 3d + 1 4 3d − 3 1 Rotated-surface [35] d × d d2 d2 − 1 1 Surface [33] d × d 2d2 − 2d + 1 2d2 − 2d 1 Toric [32] d × d 2d2 2d2 − 2 2 ∗ for triangular colour codes the dimension is defined by the side length of the equilateral triangle

−3 −2 −3 −2 pth = 2.61 × 10 pth = 1.54 × 10 pth = 2.71 × 10 pth = 2.04 × 10 100 100

10−1 10−1

10−2 d = {3, 5, 7, 9, 11} 10−2 d = {3, 5, 7, 9, 11} r 1 , 1 , 1 , 1 , 1 1 1 1 1 1 Q = ) 7 19 37 61 91 * rQ = ) 9 , 25 , 49 , 81 , 121 * 10−3 10−3

10−4 10−4 QBER QBER 10−5 10−5

10−6 10−6

− Unprotected Hadamard − Unprotected Hadamard 10 7 10 7 Upper Bound Upper Bound Lower Bound Lower Bound 10−8 10−8 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 Depolarizing probability (p) Depolarizing probability (p)

(a) Colour codes (b) Rotated-surface codes

−3 −2 −3 −2 pth = 1.26 × 10 pth = 2.13 × 10 pth = 1.36 × 10 pth = 2.17 × 10 100 100

10−1 10−1

d = {3, 5, 7, 9, 11} 10−2 10−2 d = {3, 5, 7, 9, 11} 1 1 1 1 1 rQ = ) , , , , * 1 1 1 1 1 9 25 49 81 121 rQ = ) 13 , 41 , 85 , 145 , 221 * 10−3 10−3

10−4 10−4 QBER QBER 10−5 10−5

10−6 10−6

− Unprotected Hadamard − Unprotected Hadamard 10 7 10 7 Upper Bound Upper Bound Lower Bound Lower Bound 10−8 10−8 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 Depolarizing probability (p) Depolarizing probability (p)

(c) Surface codes (d) Toric codes Fig. 16: Upper-bound and lower-bound analytical QBER performance curves of the transversal implementation of Hadamard gates protected by QTECCs.

50 syndrome vectors are ignored, because making a wrong In order to provide a confidence interval for the QBER decision, for instance by making a random decision for the performance of the QSCs for protecting single-qubit quan- error recovery operator, will further degrade the quantum state tum gates, we characterize the lower-bound performances of of the physical qubits instead of improving it. the CSS-type QSCs based on the classical sphere-packing 22 bound [83], [84], which is also known as the quantum Gilbert- minimum distance of the code d, we simultaneously increase Varshamov (GV) bound [5], [60]. Formally, the lower-bound the number of the pysical qubits length n and decrease the of the analytical QBER performance can be defined as fol- quantum coding rate rQ [38]. The quantum coding rate rQ lows [83], [84]: portrayed in Fig. 16 is calculated using Eq. (26), where the

t0+1 number of logical qubits k and the number of physical qubits QBER(1) (n, d, p) =1 A(i)pi(1 p)n−i, (82) n are also given in Table II. lower − − i=0 X Increasing the error correction capability of QTECCs means where for the QSCs-protected tranversal single-qubit gates we that we simultaneously increase the number of auxiliary qubits have p = pf and A(i) is a positive integer obeying and reduce the quantum coding rate rQ. On the other hand, at high error rates, even powerful QECCs having a lower t0+1 (n−k)/2 n quantum coding rate often carry out flawed corrections, hence Ai 2 , where Ai = . (83) ≤ i actually degrading the QBER more at high depolarizing prob- i=0   X ability values than their higher-rate counterparts. The superior Consequently, the final coefficient At0+1 is given by error correction capability of the lower rate quantum codes t0 start to impose below a specific depolarizing probability p, (n−k)/2 At0+1 = 2 Ai. (84) which we refer to as the depolarizing probability threshold − i=0 (p ). More specifically, p represents the point below which X th th If the final coefficient is At0+1 = 0, the code construction is increasing the error correction capability of a quantum code referred to as a perfect CSS-type QSC. Otherwhise, the code is considered to be beneficial. Additionally, the value of pth is referred to as a quasi-perfect CSS-type QSC. also can be used to infer the asymptotic performance of the For instance, based on Eq. (81), the upper-bound of analyt- QTECCs exhibiting a large number of physical qubits, where ical QBER performance of a distance-3 surface code can be the quantum coding rate rQ aprroches zero. In Fig. 16, the approximated as pth is denoted by a dashed line at the cross-over point of the QBER performance curves. First, we obtain the upper- (1) 13 12 QBER (13, 3, pf ) = 1 (1 pf ) 13pf (1 pf ) . upper − − − − bound of pth for colour codes, rotated-surface codes, surface (85) codes, and toric codes as follows: 2.61 10−3, 2.71 10−3, −3 −3 × × By comparison, based on Eq. (82), the lower-bound of analyt- 1.26 10 , and 1.36 10 , respectively. These specific × × ical QBER performance of a distance-3 surface code can be pth values are obtained by taking into account the errorneous approximated as stabilizer measurements. Specifically, each of the physical qubits of rotated-surface and surface codes experience only (1) 13 12 QBERlower(13, 3, pf ) = 1 (1 pf ) 13p(1 pf ) at most two X and two Z stabilizer measurements, while −2 − 11− − 50pf (1 pf ) . (86) for colour codes, each of the physical qubits experience at − − most only three X and three Z stabilizer measurements. These The analytical upper-bound QBER performance of Eq. (81) numbers of the stabilizer measurements experienced by each and the lower-bound of Eq. (82), as well as the simulation re- of the physical qubits are independent to the total number sults for the transversal Hadamard gates protected by selected of the physical qubits. In a case where we want to consider distance-3 QTECCs are depicted in Fig. 15. In Fig. 15(a), we that we have perfect stabilizer measurements, indeed we can observe that the upper-bound analytical QBER performance achieve a higher p values. We can simply use Eq. (75) of Eq. (81) is a very good approximation for the QBER th and (78) to obtain the p values associated with error-free performance from the simulation results. It is due to the fact th stabilizer measurements. More specifically, we obtain the p that the construction of distance-3 colour code is identical with th values as follows: 1.83 10−2, 1.36 10−2, 6.30 10−3, and the 7-qubit Steane’s code, which is a non-degenerate QSC. In 6.80 10−3, respectively,× for colour,× rotated-surface,× surface, general, Eq. (81) is a good approximation for non-degenerate and toric× codes. QSCs. By contrast, a slightly different phenomenon can be obseved in Fig. 15(b) and 15(c), where the simulation results Secondly, in Fig. 16, we can also observe the lower-bound of −2 −2 −2 are closer with the lower-bound analytical QBER performance the pth values given by: 1.54 10 , 2.04 10 , 2.13 10 , × × × owing to their highly degenerate property. and 2.17 10−2, respectively, for colour, rotated-surface, sur- × Next, in Fig. 16, we portray the analytical upper-bound and face, and toric codes, which are associated with errorneous sta- the lower-bound QBER performance of transversal Hadamard bilizer measurements. Similarly, when we assume that we have gates protected by QTECCs for various minimum distance error-free stabilizer measurements, the pth values obtained values. More specifically, we calculate the upper-bound QBER are: 10.78%, 10.20%, 10.65%, and 10.85%, respectively, for performance of colour codes, rotated-surface codes, surface colour, rotated-surface, surface, and toric codes. We observe codes, and toric codes having the minimum distances of that all the pth lower bound values are in the proximity of d = 3, 5, 7, 9, 11 using Eq. (81) and their lower-bound pth 11%, which is close to the quantum hashing bound for ≈ QBER{ performances} using Eq. (82). The code parameters of dual-containing CSS-type QSCs, formulated as: the QTECCs used for calculating the upper-bound and lower- CQ = 1 2H(p), (87) bound QBER performance are summarized in Table II. One of − the unique properties of QTECCs is that upon increasing the where H(p) is the binary entropy of p defined by H(p) = 23

p log p (1 p) log (1 p). Since the quantum coding two-qubit quantum gates, which is denoted as QBER(2) , can − 2 − − 2 − upper rate of two-dimensional QTECCs tends to zero (rQ 0) for be determined by modifying Eq. (81) as follows: → a large n (n ), we find that in the asymptotical limit, 2 (2) (1) the QBER performance→ ∞ of QTECCs approaches the ultimate QBER (n, d, p) = 1 1 QBER (n, d, p) upper − − upper hashing limit of dual-containing CSS-type QSCs, which is  d−1  2 ∗ t=b 2 c given by p = 11%, as we have rQ = 0. It is important to note n = 1 pi(1 p)n−i . that the quantum GV bound is relaxed optimistic lower-bound −  i −  i=0 for non-degenerate QSCs. We also have to mention that the X   value p∗ = 11% is also obtained in a different way in [82].  (90) Finally, we present the performance of the transversal In this case, we assume that p = pf considering the errorneous single-qubit quantum ggates protected by QSCs in terms of stabilizer measurements, which is given by Eq. (75) for the fidelity (Fth) of each of the single-qubit quantum gate. rotated-surface codes, surface codes, and toric codes, and by Based on Eq. (64), we define the output fidelity as follows: Eq. (78). Similar to the case of transversal Hadamard gates, we also Fout = 1 QBER . (88) − protected incorporate the formula of Eq. (82) for determining the lower- Since for asymptotical limit the QBER performance ap- bound of the analytical QBER performance of the transversal proaches the pth value, we can also determine the fidelity CNOT gates protected by QSCs as follows: threshold (Fth), which is defined as the minimum fidelity 2 required for each of the quantum gates in order to benefit QBER(2) (n, d, p) = 1 1 QBER(1) (n, d, p) upper − − lower from employing the transversal quantum gates protected by the 2  t0+1  QSCs. Explicitly, the value of Fth for a single-qubit quantum = 1 1 A(i)pi(1 p)n−i gate can be simply determined as follows: −  − −  i=0 X Fth = 1 pth. (89)  (91) − Based on all the results presented in Fig. 16, we can obtain given that A(i) is a positive integer coefficient obeying the upper-bound of the Fth for single-qubit quantum Clifford t0+1 gates as follows: 99.74%, 99.73%, 99.87%, and 99.86%, (n−k)/2 n Ai 2 , where Ai = . (92) respectively, for colour codes, rotated-surface codes, surface ≤ i i=0   codes, and toric codes. Similarly, we also obtain the lower- X Consequently, the final coefficient At0+1 is determined by bound of the Fth for single-qubit quantum Clifford gates from Fig. 16 as follows: 98.48%, 97.96%, 97.87%, and 97.83%, t0 (n−k)/2 respectively, for colour codes, rotated-surface codes, surface At0+1 = 2 Ai. (93) − codes, and toric codes. These values mark the minimum i=0 Fth X requirement for the physical implementation of the single- In order to verify the analytical expression of Eq. (90), qubit quantum gates if the implementation of QECCs within we have simulated the performance of QTECC-protected the quantum computers is considered. We have demonstrated transversal CNOT gates protected by a distance-3 colour code, that any quantum gates exhibiting lower Fth values will not rotated-surface code, and surface code, exploiting the classical- provide any benefit of reliability improvement offered by the to-quantum Pauli isomorphism. The results are depicted in QECCs. Finally, all the results presented in this subsection are Fig. 17. As expected, a similar trend to the QBER performance summarized in Table III. to the transversal Hadamard gates protected by QTECCs is displayed. An improvement in terms of QBER can be observed for depolarizing probabilities p < 2 10−3. Therefore, we C. QTECC-Protected Transversal CNOT Gates conclude that our design in Fig. 6(b)× indeed improves the Following similar investigations to those in Subsec- reliability of both Hadamard gates as well as of CNOT gates. tion VI-B, we also employed the family of QTECCs for Furthermore, we have plotted the upper-bound QBER an- protecting the transversal CNOT gates. For the transversal alytical performance given in Eq. (90) for the transversal configuration of CNOT gates, we have two sets of QSCs, one implementation of CNOT gates employing various types of QSC is invoked for protecting the physical control qubits and QTECCs for different minimum distances d based on the code another one is for the physical target qubits. We assumed the parameters of Table II. The corresponding QBER performance QSCs used for both target and control qubits are identical. curves are portrayed in Fig. 18. We can also infer pth values Due to the stabilizer preservation of QSCs after the transversal from Fig. 18, which defines the depolarizing probability value CNOT gates, as demonstrated in Section IV, each of the where we can start observing QBER improvements upon QSCs handles the errors inherent in the target qubits and in decreasing the quantum coding rate. Again, it also marks the the control qubits independently. The scheme considered as QBER performance of the QSCs at the asymptotical limit. The success if both QSCs for control and target qubits perform a pth is represented by dashed lines in Fig. 18, where we have flawless error correction procedure simultaneously. Therefore, 2.61 10−3, 2.71 10−3, 1.26 10−3, and 1.36 10−3, the upper-bound QBER performance of the QSC-protected respectively,× for colour× codes, rotated-surface× codes,× surface 24

codes, and toric codes. Similarly, we have also plotted the −2 10 lower-bound analytical QBER performance of the transversal CNOT gates protected by various QTECCs in Fig. 18 based on −2 Eq. (91). In this case, we have the pth values of 1.54 10 , −3 10 2.04 10−2, 2.13 10−2, and 2.17 10−2, respectively,× for colour,× rotated-surface,× surface, and toric× codes. Since the improvement in QBER domain can be clearly −4 10 observed in Fig. 18, it is logical that the corresponding fidelity QBER improvement can also be achieved. The essential question is what level of the fidelity the quantum gates have to be provided 10−5 Unprotected CNOT gate (simulation) Unprotected CNOT gate (analytical) for ensuring that the QSC-protected quantum gates improve Colour code (rQ = 1/7) (simulation) the overall fidelity of the system. Firstly, we have defined the Colour code (upper bound) Colour code (lower bound) output fidelity of the CNOT gates in Eq. (68), which is given 10−6 0 0.5 1 1.5 2 below: Depolarizing probability (p) ×10−3 Fout = 1 QBER . (94) − protected (a) Colour code. For asymptotical limit, based on Eq. (68), we can determine − 10 2 the threshold fidelity (Fth) as follows: 2 Fth = 1 2pth + p − th −3 1 2pth, (95) 10 ≈ − for pth 1. First, based on the upper-bound analytical QBER performance curves portrayed in Fig. 17, we can obtain the 10−4 upper-bound Fth values required by each of the CNOT gates QBER in order to gain the benefit of employing QSCs for protecting the transversal CNOT gates. By using Eq. (95), we have the 10−5 Unprotected CNOT gate (simulation) Unprotected CNOT gate (analytical) upper-bound of the Fth values of 99.48%, 99.46%, 99.74%, Rotated-surface code (rQ = 1/9) (simulation) and 99.72%, respectively, for colour codes, rotated-surface Rotated-surface code (upper bound) Rotated-surface code (lower bound) codes, surface codes, and toric codes. Additionally, based on 10−6 0 0.5 1 1.5 2 Fig. 17, the lower-bound of the Fth can also be obtained − Depolarizing probability (p) ×10 3 using Eq. (95), where we have: 99.48%, 99.46%, 99.74%, and (b) Rotated-surface code. 99.72%,, respectively, for colour codes, rotated-surface codes, surface codes, and toric codes. All of the results presented in 10−2 this subsection are summarized in Table III. Finally, we have shown that our framework proposed for protecting transversal quantum Clifford gates will provide a reliability improvement 10−3 for both single-qubit and two-qubit quantum gates, provided that the fidelity threshold required for each of the quantum gates is satisfied. 10−4 QBER VII.CONCLUSIONSAND FUTURE WORKS

10−5 Unprotected CNOT gate (simulation) A. Conclusions Unprotected CNOT gate (analytical) Surface code (rQ = 1/13) (simulation) We have presented a general framework for protecting Surface code (upper bound) Surface code (lower bound) quantum Clifford gates using QSCs along with the notion 10−6 0 0.5 1 1.5 2 of the effective stabilizer formalism. In this treatise, we Depolarizing probability (p) ×10−3 have also provided examples on how to utilize the advo- (c) Surface code. cated framework for protecting quantum Clifford gates. More specifically, in order to protect quantum Clifford gates, we Fig. 17: QBER performance for transversal configuration of arrange the gates in a transversal configuration in order to CNOT gates protected by distance-3 QTECCs over quantum exploit the benefit of stabilizer operator preservation. Fur- depolarizing channel. The quantum decoherence is inflicted thermore, since we considered imperfect quantum gates and after the transversal configuration of CNOT gates, where the c also imperfect stabilizer measurements in our scheme, we error operators of i=1 Pi n are defined by pi = p. The invoked the QTECCs for correcting the errorneous physical number of consecutive error∈ operators P c = 7 for colour codes Q qubits. The additional benefit of employing QTECCs is that the and c = 5 for surface and rotated-surface codes. number of stabilizer measurements experienced by each of the physical qubits remains constant, as we increase the number of physical qubits. Hence, the spreading and propagation of 25

−3 × −2 −3 −2 pth = 2.61 × 10 pth = 1.54 10 pth = 2.71 × 10 pth = 2.04 × 10 100 100

10−1 10−1

d , , , , − = {3 5 7 9 11} − d , , , , 10 2 10 2 = {3 5 7 9 11} 1 1 1 1 1 rQ = ) , , , , * 1 1 1 1 1 7 19 37 61 91 rQ = ) 9 , 25 , 49 , 81 , 121 *

10−3 10−3

10−4 10−4 QBER QBER 10−5 10−5

10−6 10−6

− Unprotected CNOT − Unprotected CNOT 10 7 10 7 Upper Bound Upper Bound Lower Bound Lower Bound 10−8 10−8 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 Depolarizing probability (p) Depolarizing probability (p)

(a) Colour codes (b) Rotated-surface codes

−3 −2 −3 −2 pth = 1.26 × 10 pth = 2.13 × 10 pth = 1.36 × 10 pth = 2.17 × 10 100 100

10−1 10−1 d = {3, 5, 7, 9, 11} d = {3, 5, 7, 9, 11} 1 1 1 1 1 − − rQ = ) , , , , * 10 2 1 1 1 1 1 10 2 9 25 49 81 121 rQ = ) 13 , 41 , 85 , 145 , 221 *

10−3 10−3

10−4 10−4 QBER QBER 10−5 10−5

10−6 10−6

− Unprotected CNOT − Unprotected CNOT 10 7 10 7 Upper Bound Upper Bound Lower Bound Lower Bound 10−8 10−8 10−4 10−3 10−2 10−1 10−4 10−3 10−2 10−1 Depolarizing probability (p) Depolarizing probability (p)

(c) Surface codes (d) Toric codes Fig. 18: Upper-bound and lower-bound analytical QBER performance curves of the transversal implementation of CNOT gates protected by QTECCs.

TABLE III: The summary of results of QTECC-protected transversal quantum Clifford gates.

Results Colour codes Rotated-surface codes Surface codes Toric codes

Upper bound pth (error-free measurement) 1.83% 1.36% 0.63% 0.68%

Lower bound pth (error-free measurement) 10.78% 10.20% 10.65% 10.85%

Upper bound pth (with errorneous measurement) 0.26% 0.27% 0.13% 0.14%

Lower bound pth (with errorneous measurement) 1.54% 2.04% 2.13% 2.17%

Upper bound Fth for Hadamard gates 99.74% 99.73% 99.87% 99.86%

Lower bound Fth for Hadamard gates 98.46% 97.96% 97.87% 97.83%

Upper bound Fth for CNOT gates 99.48% 99.46% 99.74% 99.72%

Lower bound Fth for CNOT gates 96.92% 95.92% 95.74% 95.66% decoherence due to the interaction between qubits during of the analytical QBER performance for both single-qubit stabilizer measurements can be effectively mitigated. We have and two-qubit transversal quantum gates protected by QSCs. shown that by combining the transversal implementation of Based on the upper-bound of the QBER performance, we quantum Clifford gates and the QTECCs, we can indeed obtain the critical Fth values for single-qubit quantum gates, improve the reliability of quantum Clifford gates, provided which are as follows: 99.74%, 99.73%, 99.87%, and 99.86%, that they satisfy a minimum depolarization fidelity threshold when protected by the colour, rotated-surface, surface, and Fth. In order to determine the approximate value of Fth, toric codes, respectively. These Fth values were obtained at first we provide both the upper-bound and the lower-bound the asymptotical limit, namely when the number of physical 26 qubits is extremely large (n ) and the quantum coding create the universal set of quantum gates for quantum com- → ∞ rate tends to zero (rQ 0). We need to emphasize that putation. Ultimately, our final goal is to construct a universal the upper-bound analytical→ QBER performance represents the framework of protecting large-scale quantum computers using worst-case scenario, where the error correction performance is QECCs. purely defined by the minimum distance of the QSCs, whilst ignoring the potential advantage of exploiting the degeneracy REFERENCES property inhereted by QSCs. As a comparison, based on [1] D. Gottesman, Stabilizer codes and quantum error correction. PhD the lower-bound of the analytical QBER performance, we thesis, California Institute of Technology, 1997. [2] P. W. 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Daryus Chandra (S’15) received the M.Eng. degree Panagiotis Botsinis (S’12-M’16) received the in electrical engineering from Universitas Gadjah M.Eng. degree from the School of Electrical and Mada, Indonesia, in 2014. He is currently pursu- Computer Engineering of the National Technical ing the Ph.D. degree with the Next Generation University of Athens (NTUA), Greece, in 2010, as Wireless ResearchGroup, School of Electronics and well as the M.Sc. degree with distinction and the Computer Science, University of Southampton, UK. Ph.D. degree in Wireless Communications from the He is a recipient of scholarship award from the University of Southampton, UK, in 2011 and 2015, Indonesia Endowment Fund for Education (Lembaga respectively. He is currently working as a Research Pengelola Dana Pendidikan, LPDP). Fellow in the Southampton Wireless group at the His research interests include classical and quan- School of Electronics and Computer Science of tum error correction codes, quantum information, the University of Southampton, UK. Since October and quantum communications. 2010, he has been a member of the Technical Chamber of Greece. His research interests include quantum-assisted communications, quantum computation, iterative detection, OFDM, MIMO, multiple access systems, coded modulation, channel coding, cooperative communications, as well as combinatorial optimization.

Zunaira Babar received her B.Eng. degree in elec- trical engineering from the National University of Soon Xin Ng (S’99-M’03-SM’08) received the Science & Technology (NUST), Islamabad, Pak- B.Eng. degree (First class) in electronic engineering istan, in 2008, and the M.Sc. degree (Distinction) and the Ph.D. degree in telecommunications from and the Ph.D degree in wireless communications the University of Southampton, Southampton, U.K., from the University of Southampton, UK, in 2011 in 1999 and 2002, respectively. From 2003 to 2006, and 2015, respectively. he was a postdoctoral research fellow working on Her research interests include quantum error cor- collaborative European research projects known as rection codes, channel coding, coded modulation, SCOUT, NEWCOM and PHOENIX. Since August iterative detection and cooperative communications. 2006, he has been a member of academic staff in the School of Electronics and Computer Science, University of Southampton. He is involved in the OPTIMIX and CONCERTO European projects as well as the IU-ATC and UC4G projects. He is currently an Associate Professor in telecommunications at the University of Southampton. His research interests include adaptive coded modulation, coded modula- tion, channel coding, space-time coding, joint source and channel coding, iterative detection, OFDM, MIMO, cooperative communications, distributed Hung Viet Nguyen received the B.Eng. degree in coding, quantum error correction codes and joint wireless-and-optical-fibre Electronics & Telecommunications from Hanoi Uni- communications. He has published over 200 papers and co-authored two John versity of Science and Technology (HUST), Hanoi, Wiley/IEEE Press books in this field. He is a Senior Member of the IEEE, Vietnam, in 1999, the M.Eng. in Telecommunica- a Chartered Engineer and a Fellow of the Higher Education Academy in the tions from Asian Institute of Technology (AIT), UK. Bangkok, Thailand, in 2002 and the Ph.D. degree in wireless communications from the University of Southampton, Southampton, U.K., in 2013. Since 1999 he has been a lecturer at the Post & Telecom- munications Institute of Technology (PTIT), Viet- nam. He is involved in the OPTIMIX and CON- Lajos Hanzo (M’91-SM’92-F’04) received his de- CERTO European projects. He is currently a postdoctoral researcher at gree in electronics in 1976 and his doctorate in Southampton Wireless (SW) group, University of Southampton, UK. 1983. In 2009 he was awarded the honorary doc- His research interests include cooperative communications, channel coding, torate “Doctor Honoris Causa” by the Technical network coding, and quantum communications. University of Budapest. During his 38-year career in telecommunications he has held various research and academic posts in Hungary, Germany and the UK. Since 1986 he has been with the School of Electronics and Computer Science, University of Southampton, UK, where he holds the chair in telecommunications. He has successfully supervised 112 PhD students, co-authored 18 John Wiley/IEEE Press books on mobile Dimitrios Alanis (S’13) received the M.Eng. degree radio communications totalling in excess of 10 000 pages, published 1692 in Electrical and Computer Engineering from the research contributions at IEEE Xplore, acted both as TPC and General Chair of Aristotle University of Thessaloniki in 2011 and the IEEE conferences, presented keynote lectures and has been awarded a number M.Sc. and PhD degrees in Wireless Communications of distinctions. Currently he is directing a 100-strong academic research team, from the University of Southampton in 2012 and working on a range of research projects in the field of wireless multimedia 2017, respectively. He is currently working as a communications sponsored by industry, the Engineering and Physical Sciences Research Fellow in Southampton Wireless (SW) Research Council (EPSRC) UK, the European Research Councils Advanced group, School of Electronics and Computer Science Fellow Grant and the Royal Societys Wolfson Research Merit Award. He is of the University of Southampton, UK. an enthusiastic supporter of industrial and academic liaison and he offers a His research interests include quantum compu- range of industrial courses. tation and quantum information theory, quantum Lajos is a Fellow of the Royal Academy of Engineering, of the Institution search algorithms, cooperative communications, resource allocation for self- of Engineering and Technology, and of the European Association for Signal organizing networks, bio-inspired optimization algorithms and classical and Processing. He is also a Governor of the IEEE VTS. During 2008–2012 quantum game theory. he was the Editor-in-Chief of the IEEE Press and a Chaired Professor also at Tsinghua University, Beijing. He has 30 000+ citations. For further information on research in progress and associated publications please refer to http://www.wireless.ecs.soton.ac.uk.