Method for constructing a quantum error correction code, a quantum processing device for implementing a quantum error correction code, a quantum error correction code and a method for implementing quantum information using the quantum error correction code

Abstract

A method for constructing a quantum error correction code with a code length n', wherein said quantum error correction code is an extended code constructed from a base code, the base code having a code length n being smaller than the code length of the extended code n', said base code requiring a base code connectivity between qubits of a first set of data and syndrome qubits, wherein said extended code requires a first connectivity between qubits of said first set, a second connectivity between qubits of a second set of data qubits and syndrome qubits, said first and second sets being different from each other, and a linking connectivity for connecting qubits in the two different sets, wherein said first connectivity is at most the base code connectivity.

Description

[0001] Method for constructing a quantum error correction code, a quantum processing device for implementing a quantum error correction code, a quantum error correction code and a method for implementing quantum information using the quantum error correction code

[0002] The present invention is related to a method for constructing a quantum error correction code with a code length n', to a quantum processing device for implementing a quantum error correction code, to a quantum error correction code and to a method for implementing quantum information using the quantum error correction code.

[0003] Quantum information is prone to destructive errors while stored or manipulated. The number of errors increases with the number of components in a quantum circuit. Therefore, active quantum error correction (QEC) is deemed essential to realize scalable quantum computer architectures [1-3]. A QEC code encodes physical quantum data in a larger space in order to introduce redundancy, so that errors on few physical qubits do not alter logical information irreversibly. Errors on the logical state are detected via a series of non-destructive, projective measurements, called parity checks, on a plurality of qubits thereby yielding a syndrome. The syndrome is processed via a decoder, tasked with determining the optimal recovery operation which restores the error-free logical state. Therefore, apart from data qubits that encode computational information, one requires auxiliary, syndrome qubits to perform the parity checks. Generally, each parity check is associated with a syndrome qubit for use in the measurement of the associated parity check. The syndrome qubit may store the syndrome. The full sequence of syndrome measurement, syndrome decoding and recovery comprise a QEC cycle and requires huge spacetime computational overheads [2, 4],

[0004] A code can generally be defined via its parity-check matrix, where each row represents a syndrome qubit and each column represents a data qubit. Equivalently, it can be represented by a bipartite “factor” graph [6] which illustrates the connections as edges between vertices that represent syndrome qubits and data qubits.

[0005] Ideally, a QEC code is implemented on a quantum processing unit (QPU) which is designed such that the connectivity between its data qubits and its syndrome qubits reflects the structure of the parity checks of the code. In particular, it is desired that no SWAP gates are required for the implementation of the code, and that the implementation of the QEC code requires a circuit with a reduced depth. Thus, it is desired that each syndrome qubit is connected to the data qubits on which the respective parity check acts non-trivially. I.e., it is desired that the connectivity provided by the QPU includes the connectivity of the code. A common problem in QEC is to come up with codes that allow for efficient encoding and decoding of information and that have a connectivity realizable by state-of-the-art QPLIs.

[0006] Low Density Parity-Check (LDPC) codes [5] are classical codes requiring low connectivity, as the number of data qubits per parity-check and the number of parity-checks that a data qubit participates in are both bounded by a constant, independently of the scaling of the code. This is represented by a sparse parity-check matrix. LDPC codes form very successful error-correction protocols as they satisfy the sparsity condition and they also saturate upper bounds on the amount of information that can be reliably transferred through noisy channels [5, 7]. As a result, they constitute the industrial standard for many modern technologies such as the WiFi and 5G [8, 9],

[0007] Quantum analogues of LDPC codes have recently attracted a lot of attention, with many families of quantum LDPC (qLDPC) codes [10-12] being developed. There are different methods of constructing quantum LDPC codes. One approach is to generate a quantum code with parity-check matrix H via the hypergraph product construction

[0013] which allows any classical code to be converted to a quantum code. This approach permits the construction of random quantum LDPC codes, which may generally possess favourable properties but may also require unrestricted qubit connectivity. Furthermore, there is no methodical way of scaling these codes, i.e. , designing larger codes from smaller ones, as their parity-check matrices are in general unrelated.

[0008] On the other hand, topological codes [14-16], such as the surface codes, are quantum LDPC codes that arise as the hypergraph product of classical repetition codes. They possess some of the highest known noise thresholds and are local in the sense that all parity checks can be performed with four stabilizer measurements on nearest-neighbour qubits

[0017] allowing for a good practical realization on a QPU. However, they provide a vanishing logical qubit encoding rate as the size increases. Semi-topological codes

[0018] have been constructed via a process that augments the edges on the factor graph of the underlying classical LDPC code that defines a given quantum LDPC code via the hypergraph product. This allows for a tradeoff between code threshold and parity-check locality.

[0009] Recently, quantum LDPC codes have demonstrated rapid success in the asymptotic regime. Good quantum LDPC codes of code length n with a non-vanishing encoding rate and a minimum distance d linear in the code length n have been discovered [19, 20], It is well- known that these good properties come at the expense of a number of long-range interactions [21-23] which complicates their practical realization on state-of-the-art QPUs. Significant progress is made in providing remedies for the QEC cycle of quantum LDPC codes in terms of guaranteeing the existence of a syndrome extraction circuit [24, 25] and fast decoders

[0026] , However, these asymptotically good constructions do not yet guide us towards practical implementations of quantum LDPC codes with small code lengths n.

[0010] Due to these problems in the prior art, it is therefore an object of the present invention to provide a method for constructing a quantum error correction code which is well-suited for an implementation on state-of-the-art quantum processing devices and to provide a quantum processing device for carrying out said method.

[0011] According to a first aspect of the present invention, there is provided a method for constructing a quantum error correction code with a code length n’, wherein said code is an extended code constructed from a base code, the base code having code length n being smaller than the code length of the extended code n’, said base code requiring a base code connectivity between qubits of a first set of data and syndrome qubits, wherein said extended code requires a first connectivity between qubits of said first set, a second connectivity between qubits of a second set of data qubits and syndrome qubits, said first and second sets being different from each other, and a linking connectivity for connecting qubits in the two different sets, wherein said first connectivity is at most the base code connectivity.

[0012] The present application was originally filed partially in color, and any reference to color in the present copy refers to the colors as originally filed.

[0013] A quantum error correction code may be specified by three numbers: Its code length n, its dimension k, and its distance d. The code may be called a [[n, k, d]] code. Such a code uses n data qubits to encode 2k(logical) quantum states, d physical errors are required to cause a logical error. The base code is a code of length n. Le., the base code uses n data qubits for the encoding. The implementation of the base code may require the implementation of Cs.i parity checks which are non-destructive projective measurements. In one example, the implementation of each parity check requires a syndrome qubit, in particular one syndrome qubit. In one embodiment, cs.i = n-k. In another embodiment, cs,i = n. In particular, the number of syndrome qubits may depend on the number of parity checks. In particular, each parity check of the base code may be associated with a respective syndrome qubit, and in particular with one respective syndrome qubit. Thus, the first set of qubits may comprise n data qubits and a first number cs,i of syndrome qubits corresponding to the Cs,i parity checks of the base code and for use of their implementation. The extended code is a code of length n’. I.e., the extended code uses n’ data qubits for the encoding of information. The implementation of the extended code may require the implementation of cs parity checks. In one example, the implementation of each parity check requires a syndrome qubit, in particular one syndrome qubit. In one embodiment, cs = n-k. In another embodiment, cs = n. In particular, the number of syndrome qubits may depend on the number of parity checks. Therefore, the implementation of the extended code may require cs syndrome qubits for use in the measurement of the parity checks of the extended code. In particular, each parity check of the extended code may be associated with a respective syndrome qubit, and in particular with one respective syndrome qubit.

[0014] The n’ data qubits and the cs syndrome qubits may be divided into the first set of data and syndrome qubits and the second set of data and syndrome qubits. The first set of qubits may comprise n data qubits and a first number cs.i of syndrome qubits corresponding to the Cs,i parity checks of the base code and for use of their implementation. The second set of qubits may comprise n2 = n’ - n data qubits and C2 = cs - Cs,i syndrome qubits. In one example n > n2, in another example n < n2, and in yet another example n = n2-

[0015] In one example, the implementation of the parity check associated with the syndrome qubit qsmay comprise implementing a sequence of two-qubit gates on the syndrome qubit qsand each of the data qubits in the support of the parity check associated with the syndrome qubit qsand a projective measurement of the syndrome qubit qs.

[0016] In the present invention, the qubits are not limited to a particular type and may comprise superconducting qubits, quantum dots, atoms, ions, etc.

[0017] A connectivity between two qubits allows for an interaction between them, in particular an implementation of a two-qubit entangling gate. Within a physical device, e.g., a quantum processing device, the connectivity between the qubits may be created by coupling the qubits, either directly or via a coupling structure. In the case of superconducting qubits, connectivity of the qubits may be created by direct coupling or via a resonator or wave guide. The required connectivity of the code may be understood as the entirety of connectivity between data qubits and syndrome qubits that is required to implement the quantum error correction code in the most efficient way as all connections for implementing all two-qubit gates between syndrome and data qubits according to the parity checks are present. The required connectivity of a code may thus be defined as the connectivity required by all the parity checks of the code, wherein the connectivity required by a respective parity check is the connectivity between the syndrome qubit associated with the respective parity check and the data qubits in the support of the respective parity check. The connectivity may also be represented by a graph, wherein each qubit (data or syndrome qubit) is associated with a vertex, and there is an edge between a vertex representing the syndrome qubit associated with a particular parity check and a vertex representing a data qubit when the data qubit is in the support of the particular parity check associated with the respective syndrome qubit. The support of the parity check is the space of data qubits on which the parity check acts non-trivially. A quantum processing device having the desired connectivity of the code allows for an efficient implementation of the code via an application of two-qubit gates between the syndrome and data qubits (see e.g. M. Nielsen, I. Chuang, “Quantum Computation and Quantum Information”, 10thAnniversary Edition, Chapter 10.5.8).

[0018] Within this description, the term said one connectivity required by said one code is at most said other connectivity required by said other code means that every connection between qubits according to the other connectivity is also a required connection between qubits according to the one connectivity. I.e. , the one connectivity only includes connections between qubits which are also connections of the other connectivity and does not include connections between qubits which are not connections according to the other connectivity.

[0019] The extended code is particularly suited for an implementation on a quantum processing device with a first module comprising the first set of qubits having the first connectivity, a second module comprising the second set of qubits having the second connectivity, and the linking connectivity between the qubits of the first and second sets. The first and second sets are different from each other. In particular, there is no qubit of the first (second) set which is also a member of the second (first) set in one example. In this case we say that the first and second sets are disjoint meaning they are different from each other. The connectivity is in particular a connectivity between each of a plurality of syndrome qubits and a respective plurality of data qubits associated to said syndrome qubit. The first and second modules may be physically separated entities in one example, e.g., two separated chips. In another example, the first and second modules may refer to different areas of a quantum processing device comprising the data and syndrome qubits of the first and second sets.

[0020] In one example, the qubits of the first module may have the base code connectivity. In this case, the base code and the extended code may be implemented efficiently on the same quantum processing device.

[0021] In one embodiment, the extended code is a non-local code. I.e., the extended code comprises stabilizer operators that are non-local, see e.g. H. Bombin, “An introduction to Topological Quantum Codes”, arxiv: 1311.0277v1. In particular, the base code and / or the extended code is not a topological code. In one example, the base code and / or the extended code is not a Toric code or is not a surface code or is not a color code.

[0022] In one embodiment of the method, the extended code may further require for at least one module set of data and syndrome qubits of said second set a module connectivity between qubits of said at least one module set, said module connectivity being at most the base code connectivity. In one example, the extended code may require for each of a plurality of module sets of qubits a respective module connectivity between the qubits of the respective module set. In one example, the module set may be identical to the second set. In another example, the module set may be included in the second set but may be different from the second set.

[0023] In case of more than one module set, the module connectivity of two different module sets may be identical or different from each other as long as it is at most the base code connectivity. In one example of more than one module set, the module sets may all be disjoint or different and their union may be the second set.

[0024] Furthermore, the extended code may require a respective linking connectivity between qubits in the first set and the module sets and between qubits in different module sets. In one example, there may be no qubit which is simultaneously a member of two different module sets and / or there may be no qubit which is simultaneously a member of the first set and one of the module sets. The module connectivity is in particular a connectivity between a syndrome qubit and a plurality of data qubits in the support of the parity check associated with said syndrome qubit.

[0025] When the extended code requires the module connectivity between M > 1 module sets of qubits of the second set, the extended code is particularly well suited for implementation on a quantum processing device which comprises M+1 base code modules, wherein each base code module comprises n data qubits and cs,i syndrome qubits corresponding to the cs,i parity checks of the base code, and wherein the qubits of each module set have the base code connectivity. The qubits of the different modules are connected according to the linking connectivity. Then, the quantum processing device is suitable for an efficient implementation of the base code and the extended code.

[0026] In one embodiment, the first connectivity and / or the module connectivity may be identical to the base code connectivity. Then, all connectivities required for the base code are also required for the extended code. The extended code may require further connectivities, though. In this case, a quantum processing device with the required connectivity for the implementation of the extended code also has the required connectivity or the implementation of the base code.

[0027] In one example, the extended code, and in particular also the base code, is an [[n, k, d]] quantum stabilizer code. An [[n , k, d]] quantum stabilizer code Q is defined by a stabilizer group S c y>ntgenerated by f > n-k (not necessarily independent) elements (Si,..., ), called parity checks or stabilizer generators, such that the code corresponds to the k-dimensional (+1 )- eigenspace

[0028] The set of n-qubit Pauli operators Pnis defined on the n-qubit Hilbert space H := C2" = (C2)®nand consists of all operators P = a Pi <8> ■ ■ ■ ®Pn, where a G {±1 , ±i} and P, G {I ,X, Y, Z} for i = 1 , . . . , n, wherein I, X, Y, Z are the 2x2 identity matrix, the Pauli X-matrix, the Pauli Y-matrix and the Pauli Z-matrix, respectively. The weight w of a Pauli operator P is the number of non-identity components Pi in the tensor product. The Pauli group modulo the phase factor a, i.e. Pnl{±\, ±il}, is isomorphic to the commutative group of 2n-binary strings, with the isomorphism given by ... , zn) = (x|z). It is then known that two Pauli operators P, P' G Pncommute if and only if the following condition on their respective binary representations (x, z), (x', z') G is satisfied: [P, P'] = 0 <=> x ■ z’ + z ■ x’ =

[0029] The distance d of the code indicates the number of physical errors required to cause a logical error. Formally, it is defined as the minimum weight of a Pauli operator P such that

[0030] [P, Q] = 0 for all Q G S, but P £ S.

[0031] Applying the isomorphic mapping above to the stabilizer generators Si S{, one obtains an I x 2n binary matrix H = (HX|HZ), called the parity-check matrix of Q, where the rows of the parity-check matrix correspond to the £ stabilizer generators. The null space of H consists of vectors (z|x) such that the representations (x|z) correspond to all the Pauli operators that commute with the stabilizer generators. In particular, all stabilizer generators need to commute with themselves, thus resulting in the commutativity condition HxHz + HZHX= 0 that any stabilizer code needs to satisfy. By the rank-nullity theorem, the code dimension can be calculated as k = n— rank(H), since this is the number of linearly independent codewords that H permits.

[0032] One important feature of stabilizer codes is that their structure enables systematic construction of procedures for encoding, decoding and error corrections. Quantum circuits for encoding, decoding and correction of stabilizer codes are presented in “Quantum Computation and Quantum Information”, 10thAnniversary Edition by M. Nielsen and L. Chuang, Chapter 10.5.8. Thus, the method according to the example where the extended code is a stabilizer code, allows for an efficient practical realization on a quantum computing device.

[0033] In one example, the extended code, and in particular also the base code, may be a CSS code [34, 35], defined by the property that for any stabilizer generator, the non-identity components in its tensor product representation are either all X or all Z. Mathematically, this is properly reflected by the fact that the parity-check matrix may be represented as where Hx, Hzare binary matrices with n columns for a [[n, k, d]] code. We call Hxthe X-matrix and Hzthe Z-matrix of the code in this disclosure. The commutativity condition for CSS codes therefore reads HXHZ= 0, and the code dimension can be calculated as n - rank(Hx) - rank(Hz).

[0034] In another embodiment, the extended code, and in particular also the base code, may be a CSS code, in particular quantum Low Density Parity Check code.

[0035] A classical LDPC code Qci can be thought of as the nullspace of a sparse (f x n) paritycheck matrix Hci so that every n-bit codeword c E Qci if and only if Hcrc=0. The sparsity requirement is reflected by an exponentially vanishing number of non-zero elements in the parity-check matrix as its size increases, i.e., as one considers a family of codes with paritycheck matrices Hci of increasing t and n.

[0036] Analogously, quantum LDPC codes are stabilizer CSS codes with sparse parity-check matrices. Hereon, it is assumed that the X-matrix Hxand the Z-matrix Hz of the quantum LDPC code are both (f*n) matrices, so that the parity check matrix H of the LDPC code is a (2f x 2n) parity-check matrix. We first define the row and column Hamming weights for H, which will allow us to specify a sparsity condition. The Hamming weight of row i, w^l)■= indicates the number of qubits in the i-th parity check, i.e., the number of qubits on which the i-th parity check acts non-trivially. The Hamming weight of column j, Hij indicates the number of X-parity checks that the j-th qubit participates in if 1 < j < n and the number of Z-parity checks that the (j - n)-th qubit participates in if n + 1 < j < 2n. We set wrmaxiivrwand wc■= maxj The advantage of quantum LDPC codes is that the structure of their parity checks allows for an efficient implementation on the quantum processing device as the number of connections required for the implementation of each parity check is independent of the code length.

[0037] Constructing classical LDPC codes can be as easy as sampling random parity-check matrices. In fact, this method gives classical codes with good asymptotic properties k e 0(n) and d e O(n)

[0036] . However, this simple approach generally fails for quantum LDPC codes because the commutativity constraint HxHz = 0 fails with high probability. Below, an embodiment of the method according to the present invention including a constructive way to obtain quantum LPDC codes with finite rate will be presented.

[0038] The LDPC property for quantum codes may be relaxed by introducing the notion of a q(m)-sparse code: Let T = { Qi, Q2,...} be a family of quantum CSS codes, where Qmis defined for m = 1, 2, . . . via a (2 / mx2nm) parity-check matrix Hm, with maximum row weight wr(m) and column weight wc(m) expressed as functions of m. Suppose that no row or column of Hmconsists of zeros only. Then, T is called a family of q(m)-sparse quantum codes q(m), If q(m) = O(am) for some constant a e (0, 1 ), it is said that every code in T has exponentially decaying density. If q(m)nm< q and q(m)4 < q for some constant q > 0, every code in T is called a q-LDPC (or simply LDPC) code. This definition is dependent on the representation of the parity-check matrices Hmfor m > 1 . One such representation is the standard form of the parity-check matrix, e.g. obtained by the algorithm in chapter 9.4 of Ref.

[0037] , where the matrix is full-rank and = nm- km. However, in general, the parity-check matrix may be instructed by the experimental set-up, so the definition is kept general by introducing an arbitrary fm> nm- km. It is then required that matrices Hmdo not contain any rows or columns of Hmconsisting of zeros, so that the sparsity of the matrices is not increased artificially.

[0039] For quantum LDPC codes, conditions q(m)^m< q and q(m)nm< q for constant q > 0 correspond to the statements that every parity check acts on a constant number of qubits and every qubit participates in a constant number of parity checks. Note that a q-LDPC family T of codes with nmand that scale exponentially like 0(pm) is itself a O(q / Pm)-sparse code. Therefore, quantum LDPC codes can be thought as codes with the fastest exponentially decaying density.

[0040] In one embodiment of the method, the base code may be a generalized bicycle code GB(aL, bL) of code length 2L with predetermined binary generating base code polynomials = LfJol atx'> respectively bL(x) = of degree at most L-1 > 1, i.e., the coefficients atand btare either 0 or 1 for i = 0, ... , L - 1, the base code polynomials being such that twice the degree of the greatest common divisor of the base code polynomials aL(x), bL(x) and the binary polynomial (xL-1 ) is equal to a predetermined base code dimension / c1=2deg(gcd(ctL(x), bL(x), (xL-1 )))> 2, and the extended code may be a generalized bicycle code GB(aqmL, bqmL) of code length 2qmL with predetermined binary generating polynomials “MW and bqmL(x) of degree at most qmL - 1 , which are related to the base code polynomials via aqmLM = fqm,dadx)], respectively bw(x) = / w[bL(x)], wherein is a function and qm>1 is an integer scaling factor.

[0041] A generalized bicycle code GB(a, b) (see, e.g. Ref.

[0029] ) with generating binary polynomials a(x), b(x) of degree at most £-1 is a [n=2£, k, d] CSS code defined by a parity check matrix H = wherein 0 / is an £ x I zero-matrix, A and B are £ x £ circulant matrices with predetermined binary generating polynomials respectively bf(x) = Ef=obix‘ of degree at most £ -1 , i.e., the coefficients atand btare either 0 or 1 for i = 0, 1. In the following, we may use the notations Hm, Amand Bmfor the matrices related to the extended code.

[0042] The parity check matrix H according to the above construction is indeed the parity check matrix of a CSS code since HxWj = AB + BA = [A,B] = 0, wherein Hx:= (A|B) and Hz■= (BTIAT) (i.e., (Hx)ij = Ajj analogously for Hz). When B = AT, the special case of the so-called bicycle codes

[0027] is obtained.

[0043] Random matrices Arand Brsatisfy the commutativity condition [Ar, Br] = 0 with low probability, however, the method according to the embodiment ensures the commutativity condition. Let F^ = F2[x] / denote the ring of polynomials with binary coefficients modulo x^- 1 . These are the polynomials p(x) = po + pix + • • with coefficients

[0044] Pi e {0, 1} =: F2for i = 0, . . . , -3-1 and maximum degree £ - 1 . The weight wt p(x) of a polynomial p(x) is the number of its non-zero coefficients, and gcd( • ) denotes the greatest common polynomial divisor of its arguments. The ring F^ is isomorphic to the ring of £ x £ binary circulant matrices, such that a polynomial p(x) is isomorphic to the binary circulant matrix PM, denoted by p(x) ~ PM, where The isomorphism becomes transparent when the circulant matrix is expressed as

[0045] PM = pohxf + PIXG + ' ' ' + P{~±XQ1, where W is the I x t identity matrix and XG=Efc=o |k + l><k| is the generalized Pauli X operator in I dimensions and k + 1 denotes addition modulo f. The polynomial p(x) may also be called the generating polynomial of the circulant matrix within this application.

[0046] For any two circulant matrices A and B represented by polynomials a(x) and b(x), respectively, their product AB is represented by the polynomial a(x)b(x) (mod x^-1 ). Therefore, the commutativity of A and B, hence also the commutativity condition on the paritycheck matrix H defined by A and B, is a consequence of the commutativity of polynomial multiplication over the ring F^.

[0047] Using this construction, it can be proven

[0029] that the code dimension k is related to algebraic properties of the specific polynomials a(x), b(x) that define H. In particular, the dimension of a [[2£ k, d]] code defined by a(x), b(x) e F^isgiven by k = 2 deg g(x), where g(x) = gcd(a(x), b(x), x^-1 ).

[0048] The core idea of the construction according to the above embodiment lies in defining an algebraic transformation on the generating polynomials of the base code in order to obtain an extended code with related parity-check matrix. The allowed algebraic transformations convert the generating polynomials aL(x), bL(x) E F^ of the base code into new polynomials aqmL(x) some choice of function fqm Land for some scaling factor qmwhich is chosen as a positive integer. The new parameters qmL, maydemultiples of L, 3L(X), bi_(x) on their respective rings in certain examples, allowing, at least for certain examples shown below, a guarantee that the dimension and distance of the extended code are lower bounded by the dimension and distance of the base code, respectively. The function fqm Lis chosen such that the first connectivity is at most the base code connectivity. Examples of functions which ensure this property are presented below.

[0049] Each parity check of the extended code that appears as a row in the full parity-check matrix Hmhas weight equal to the sum of Hamming weights of Amand Bm, or equivalently wr(m) each qubit appears in wc(m) = max{wt aqmL(x), wt bqmL(x)} parity checks, where wc(m) < wr(m) always. One can consider the asymptotic limit M — ► °° to study the sparsity of the parity-check matrices. One can immediately see from the above presented formulas of the Hamming weights that the above construction leads to a family of extended codes which is a family of quantum LDPC codes if and only if the functions fqm Ldo not increase the weight of the generating polynomials, i.e wr(m) < q for all m and some constant positive integer q. This can be achieved by careful choice of a(x), b(x) and / or of fqm L. Special embodiments of such families of codes will be presented below.

[0050] In one embodiment of the method, the sum of the weights of the base code polynomials aL(x) and bL(x), i.e., the sum of the number of their non-zero coefficients, wt aL(x) + wt bi(x), is at most 10, in particular at most 9, more in particular at most 8, and even more in particular at most 7. In one example, the weight of at least one of the base code polynomials czL(x) and bL(x), i.e., the number of its non-zero coefficients, and in particular the weight of each base code polynomial, wt ai(x) and / or wt bi(x) is at most 5, in particular at most 4, more in particular at most 3, and even more in particular at most 2. Extended codes constructed on the basis of such base code polynomials may result in codes with a sparse or q(m)-sparse parity check matrix or they may be qLDPC codes. The method according to the above embodiment may be efficiently implemented on a quantum processing device due to the sparsity condition.

[0051] In one embodiment, the method is such that either fqm L[aL(x)]= vqm,i(x)aL(x) and / qmL[^W]= Pqm,L &)bL(x) for some mutual polynomial pqm,LW) or wherein the function fqm Lis a non-linear function. In one example, the mutual polynomial may be defined as pqmiL(x) = ancl may be apredetermined mutual binary polynomial of degree at most (qm- 1 )L, i.e., the coefficients pt are either 0 or 1 for i = 0, ... , (qm- 1)L. Then, each extended code satisfies km> ki, i.e., the dimension of each extended code is lower bounded by the dimension fo of the base code. In some examples, dm> di is also fulfilled. I.e., the distance of the extended code is lower bounded by the distance of the base code. In this way, the extended code may have improved properties compared to the base code.

[0052] In one example, the mutual polynomial is of the form pqm,L(*) = 1 so that the extended code is a quantum Low Density Parity Check code. In this embodiment, the generating polynomials aqmL(x), and bq L(x), have the same weight independent of the code length n' = 2qmL. The extended code is thus a member of a family of codes rLDPC= {Qi, Qz, . . . }, where Qmis defined by polynomials for m = 1 , 2 which belong to rings of increasing dimension. Therefore, the circulant matrices Am, Bmare all generated by polynomials aqmL(x), bqmL(x), but they are different as m increases. In particular, the sequence (qm) m=i,2,... specifies how many additional zero elements Amand Bmcontain compared to Am^ and Bm-i, respectively. One can see that wr(m) = wt for all m = 1 , 2 which is independent of m, so TLDPCforms a family of quantum LDPC codes. For this embodiment, qubits of qm-l module sets of qubits may have the module connectivity which is at most the base code connectivity but which is different from the base code connectivity. Also the first connectivity is not identical to the base code connectivity in this embodiment.

[0053] When the mutual polynomial is of the form pqmiL(x) = 1 for each m, the following six examples of base code polynomials a(x), b(x) and base code lengths L result respectively in extended codes which are qLDPC codes with a performance that improves as the code length of the extended code increases (see also details below):

[0054] In one example, the base code polynomials are of the form ai_(x) = 1 + x4and bi(x) = 1 + x + x2+ x4, and L = 5, resulting in a [[10, 2, 3]] base code.

[0055] In another example, the base code polynomials are of the form ai(x) = 1 + x + x2+ x5and bi_(x) = 1 + x + x3+ x5, and L = 6, resulting in a [[12, 2, 3]] base code.

[0056] In one example, the base code polynomials are of the form ai(x) = 1 + x3and bi(x) = 1 + x + x3+ x6, and L = 7, resulting in a [[14, 2]] base code.

[0057] In one example, the base code polynomials are of the form ai(x) = x + x3and bu(x) = 1 + x5, and L = 8, resulting in a [[16, 2]] base code.

[0058] In one example, the base code polynomials are of the form 3L(X) = 1 + X2and bL(x) = 1 + x5, and L = 9, resulting in a [[18,2]] base code.

[0059] In one example, the base code polynomials are of the form ai_(x) = 1 + x and bi(x) = 1 + x6, and L = 10, resulting in a [[10, 2]] base code.

[0060] Properties of QEC on the basis of such extended codes will be discussed in further detail below.

[0061] In a further example, the base code polynomials are of the form ai_(x) = 1 + x7+ x8+ x9and bi(x) = 1 + x6, and L = 10. In yet another example, the base code polynomials are of the from 3L(X) = 1 + x4and bt(x) = 1 + x2+x3+ x4.

[0062] Given limitations on current quantum hardware, in particular for superconducting quantum processing devices realized on a chip, it may be important to take “scalability” of the chip into account when designing a quantum code. Assuming that a chip is designed with the required connectivity of a specific quantum code, it can then be easily replicated to create multiple identical chips or quantum memories. Once the computational capabilities need to scale, one might require a larger code to encode the logical information. In such a scenario, it may be desirable to scale the current chips without affecting the components that make up the original code, thus avoiding the need for recalibration.

[0063] The above requirement may be fulfilled by another embodiment of the method wherein the base code and the extended code are generalized bicycle codes, wherein the scaling factor is qm= 3—1for an integer with mutual binary polynomial + x3k~1 £).

[0064] Extended codes obtained in this way fulfil a strict notion of scalability in the sense that the parity check matrix Hmof any code with smaller code length nm(in particular, the base code which may have the parity check matrix Hi) may be embedded in the parity check matrix Hm' of an extended code with the larger code length nm\ That is, there exists a sequence of column and row swaps such that (H'xltj = (Hxlij and wherein H’x and H’zare the X- and Z-matrix of the code with code length nm', and Hxand Hzare the X- and Z-matrix of the code with code length nm. I.e., the first connectivity of the extended code is identical to the base code connectivity.

[0065] To be more explicit, the extended code Qmwith code length nm= 2qmL , qm= 3m'1wherein m > 2 is an integer, or the base code Qi with code length m = 2L, qi = 1 , is defined by an 2nmx 2nmparity check matrix Hm, said parity check matrix Hmbeing of the form Hm= -matrix, Amand Bmare qmL x qmL circulant matrices with predetermined binary generating polynomials aqmL(x) = pqniL(x)aL(x), respectively bqmL(x) = vqm,L(x)bL(x) of degree at most qmL - 1 for m > 2, wherein the mutual binary polynomial is of the form For m = 1 , the generating polynomials are the generating polynomials of the base code, ai_(x), bi_(x). For this family of codes, the code Qm+i is obtained from the code Qmaccording to triangular matrix obtained from Gm, respectively.

[0066] We denote by HX, HZthe X-matrix and Z-matrix of Hm>and we denote by Hx, Hz' the X- matrix and Z-matrix of Hm+i. Further, we introduce the notation Lm= L(Am), Um= U(Am} and L'm= L(Bm), = UtBm) Then, according to the above,

[0067] Hx' = By relabeling the data qubits (i.e., swapping the columns), one may arrive at

[0068] Ux, relabeled (Um\U^) This larger X-matrix embeds the smaller X-

[0069] (iml4) matrix G4m|Bm) of the code Qm. In particular, each circulant matrix Amand Bm, and thereby the X-matrix is embedded three times within the larger X-matrix, each time involving a different set of stabilizers and data qubits. The remaining bocks that make up the X-matrix of Qm+i are upper and lower triangular matrices obtained from Amand Bm. Thus, the corresponding stabilizer operators contain a strict subset of the connections between the syndrome and data qubits that appear in Amand Bm. A similar result may be obtained for Hz , the Z-matrix of Hm+i.

[0070] Thus, as the code length of the extended code grows, the data qubits remain connected to the same syndrome qubits as in the base code, up to relabelling, with the boundary data and syndrome qubits requiring new connections to newly introduced syndrome and data qubits, respectively.

[0071] While extended codes of the above embodiment are strictly scalable, they are not within the family of qLDCP codes. Indeed, one may prove that the family of codes is a family of q(m)- sparse codes defined above with q(m) = >Mwhere the polynomials « / .(%), bL(x) are the base code polynomials. I.e., the qLDPC property is weakened to the property that the parity-check matrices have exponentially decaying density.

[0072] As each circulant matrix Am, Bmis embedded three times within Am+i, Bm+i, respectively, each time involving a separate set of qubits and stabilizers, and the remaining six blocks that make up Am+i, Bm+iare lower and upper triangular matrices obtained from Am, Bm, so that they contain a strict subset of the connections between syndrome and data qubits that appear in Am, Bm, the parity check matrix of Hm+i locally shares the structure of Hm.

[0073] Le., for a scaling factor qm= 3m’1, m > 2, the extended code requires for the qubits of each of qm-1 module sets of qubits the base code connectivity. Furthermore, there is a linking connectivity between the qubits of different module set and the qubits of the module sets and the qubits of the first set. In one example, the base code polynomials ai_(x) and bi_(x) may be as in the examples presented above.

[0074] The extended codes according to the above embodiment may have improved properties with increasing code length, as the km+i> kmfor all base codes, and there are examples for which dm+i > dm.

[0075] In another embodiment of the method according to the first aspect of the present invention, the method may comprise a CSS code construction step for constructing the base code and the extended code with the following steps: i) selecting the predetermined dimension 2 < / q; ii) selecting two binary test base code polynomials = SfZo1b^x1of degree at most L-1 such that twice the degree of the greatest common divisor of the test base code polynomials h®(x) and the binary polynomial (xL-1 ) is equal to the predetermined dimension kr, i.e., 2deg(gcd (x), (xL- D))= ki; ill) constructing a CSS test base code with the base code length n-i=2L which is the generalized bicycle code GB &P), with generating test base code polynomials aff)(x), ^t}(x); iv) selecting a family of test functions f^Lfor a sequence of integer scaling factors 1 < qi < q2< < qM', v) constructing a family of extended CSS test codes with code lengths nJ = Qrn -2L which are generalized bicycle codes GB with generating test polynomials a®L(x) = fq^L[a^\x)], respectively b^x) = / ^[^(x)], of degree at most qmL - 1 ; vi) selecting a noise model for a system of m, respectively n® qubits, said noise model being described by a quantum channel EAwith an associated noise strength parameter A; vii) simulating, for a plurality of values of the noise strength parameter A, quantum error correction according the CSS test base code and the extended CSS test code on a classical computer on a model of qubits for the CSS test base code, respectively on a model of n® qubits for the extended CSS test codes, the qubits being subject to noise according to the noise model with the respective value of the noise strength parameter to thereby obtain for each value of the noise strength parameter an associated logical error rate of the respective code; viii) if for at least one extended test code with scaling factor qmthe logical error rate is below the logical error rate of the CSS test base code, selecting the test base code polynomials a®(%), b^\x) as the predetermined base code polynomials and the CSS code with scaling factor qmas the extended code of the quantum error correction method, otherwise returning to step i).

[0076] The CSS code construction step allows to find extended codes with improved performance over the base code. The CSS code construction step may be carried out by a classical computer.

[0077] In one example, the test functions may be such that fqmL and beingamutual binary polynomial.

[0078] In one example, p®L= 1. In another example, the functions fqmLmay be non-linear functions.

[0079] In one example, the noise model may be a model of unbiased depolarizing noise. I.e., a single-qubit Pauli flip ps-» P QSP occurs with probability A independently of the choice of the Pauli operator Pe {X, Y,Z}. psis the single-qubit density matrix.

[0080] According to a second aspect of the present invention, there is provided a quantum processing device for implementing a quantum error correction code constructed according to the method of anyone of the above, said quantum processing device comprising: a plurality of n’ data qubits and a plurality of syndrome qubits for implementing the extended code; characterized in that the quantum processing device comprises the first connectivity between qubits of the first set, the second connectivity between the qubits of the second set, and the linking connectivity for connecting qubits in the two different sets, wherein said first connectivity is at most the base code connectivity. The quantum processing device according to the second embodiment is particularly well- suited for the efficient implementation of the extended code as it comprises the required connectivity of the extended code at least for the qubits in the first set.

[0081] In one embodiment, the quantum processing device may include a first module comprising the first set of qubits and a second module comprising the second set of qubits. The first and second modules may be separate from each other. E.g., the qubits may be arranged on separated carriers or chips. The separate carriers or chips may be arranged on a common substrate, though. In another example, the qubits may be arranged on the same carrier or chip which is continuous in space, and the first and second sets then refer to different regions on the carrier or chip.

[0082] The qubits of the quantum processing device may comprise n' data qubits for encoding of information on the basis of the extended code and cssyndrome qubits for use in the measurement of parity checks of said extended code. In one example, the quantum processing device may comprise means for measuring the parity checks by the application of single-and two-qubit gates and local measurements on the data and syndrome qubits.

[0083] The connectivity between the qubits of the first set may be the first connectivity. In one embodiment, the connectivity of the quantum processing device may be composed of the first, second and linking connectivity. The connectivity of the quantum processing device may be identical to the required connectivity of the extended code.

[0084] In another example, the qubits of the first set may have an all-to-all connectivity. In yet another embodiment, the first set of qubits and at least a subset of the second set of qubits, e.g., one subset, may each be provided with all-to-all connectivity, and the linking connectivity provided between the qubits of the first and second sets of the quantum processing device is such that the required connectivity of the extended code is at most the connectivity of the quantum processing device. In particular, the quantum processing device may be such that there is no all-to-all connectivity between all of its qubits. In one example, the qubits are superconducting qubits, in particular transmons, and the all-to-all connectivity between qubits of a respective set is provided by a coupling structure, e.g., a wave guide or resonator.

[0085] In one example, the data and syndrome qubits may be superconducting qubits and providing the connectivity may include the use of waveguides, resonators, non-local couplers, direct coupling between the qubits, etc. In one embodiment, the connectivity between the qubits in the first set may be identical to the base code connectivity. In this case, the quantum processing device is also well-suited for the implementation of the base code as it provides the required connectivity of the base code.

[0086] In yet another example, the extended code is strictly scalable, and the quantum processing device may comprise a plurality of module sets of qubits with the same number of qubits as the first set of qubits and having the first connectivity, and in particular having the base code connectivity. The qubits of the module sets and the first set may have connectivity such that the total connectivity of all qubits is at most the required connectivity of the scalable extended code. Such a quantum processing device allows for a very efficient implementation of the base code and one or more extended codes with different code lengths. Furthermore, such a quantum processing device may allow for an efficient manufacturing by first manufacturing a plurality of modules comprising the module sets of qubits with the module connectivity and then providing a linking connectivity between the qubits of the different modules.

[0087] According to a third aspect of the present invention, there is provided a quantum error correction code, said quantum error correction code being the extended code constructed according to anyone of the above.

[0088] According to a fourth aspect of the present invention, there is provided a method of implementing quantum information using the quantum error correction code constructed according to anyone of the above, in particular for quantum error correction, quantum error mitigation and quantum metrology.

[0089] In the following, the invention will be described in more detail by way of example with reference to the drawings in which:

[0090] Figure 1 depicts logical error rates and weights for QEC on the basis of LDPC codes in families of codes constructed from base codes with small base code length;

[0091] Figure 2 depicts the performance of QEC on the basis of a family of quantum LDPC codes with base codes having a small base code length;

[0092] Figure 3 depicts the performance of QEC on the basis of codes in various families of

[0093] LDPC codes. Figure 4 schematically depicts an embodiment of a quantum processing device according to the second aspect of the present invention.

[0094] Figures 5a-5j schematically represent the required connectivities of the stabilizer operators defined by the X-matrix of a base code and an extended code according to an embodiment of the present invention.

[0095] A. Quantum error correction on the basis of extended codes which are quantum LDPC Codes

[0096] In the following, numerically obtained thresholds for QEC on the basis of base codes and extended codes which are members of families of small quantum LDPC codes according to the present invention are presented. In order to obtain the threshold values for a family of codes T, several noise simulations are run so that for each run an error string e occurs, leading to a syndrome s = H • e (syndrome equation). The goal of a decoder is to obtain the most likely error string e that satisfies this syndrome s. Decoding for the simulations was performed using the BP + OSD decoder

[0018] which we implemented via the bposd Python package

[0038] .

[0097] Belief Propagation (BP) [5, 39, 40] is the most frequently used decoder for classical LDPC codes [5], It iteratively updates the probability distribution P(e| s) of individual bits in the codeword. In the classical setting, BP aims to find the exact error string e that satisfies the syndrome equation above by updating the error string e -» e' in a sequence of “beliefs”, with the i-th bit given by rl if > 0.5, eito if < 0.5, where p(ei) = Zej*eiP(ei, ei-i, 1, ei+i, ... en|s) is the conditional probability of an error occurring on the i-th bit given the syndrome s.

[0098] In the quantum setting, there may be multiple minimum weight estimates of the error for a given syndrome, due to the stabilizer encoding, a phenomenon known as quantum degeneracy. It is then sufficient to find a recovery operation r that corrects the error e up to a stabilizer, so that r + e e rowspace (H). However, BP fails to account for quantum degeneracy as it assigns high probabilities to all minimum weight error estimates, thus failing to converge. Ordered Statistics Decoding (OSD) [29, 41] is a postprocessing algorithm supplementing BP with a lot of success for various families of LDPC codes [18, 29], The idea behind OSD is that although belief propagation does not always lead to an error estimate e that satisfies the syndrome equation (H • e * s), one can still order the set of error bits by highest likelihood of having flipped. One then extracts the string of most likely flipped bits ei with dimension equal to the column space of H. This leads to the OSD-O error estimate e = ei ® ez, with ez = 0, that always satisfies the syndrome equation. The number 0 indicates that the OSD algorithm is of order 0. In general, the OSD algorithm can be of order w, indicating a greedy search over the w most reliable bits of the remaining string ez, with higher w improving the error-correcting performance. Full details on the BP+OSD protocol and its implementation can be found in Refs [18, 29],

[0099] In the simulations, unbiased depolarizing noise is considered, i.e. a single-qubit Pauli flip p -> Pp P occurs with constant probability p independently of the choice of the Pauli operator P e {X, Y, Z}. The ‘min-sum’ variant of BP

[0042] (as described in detail in Appendix C of Ref

[0018] ) with scaling factor 0.625 and iteration depth 40 is used. The OSD order is set to w = L, where L is the dimension of the polynomial ring of the polynomials defining the base code. 5 • 104simulations are run, where each simulation is terminated at a precision of 103, a value much lower than any obtained threshold values.

[0100] In the following, examples of quantum error correction on the basis of extended codes which are quantum LDPC codes with improved performance with increasing code length, while keeping the code lengths as small as possible, are presented. In the following, 2L is the length of the base code, wherein 2 < L. 1 = qi < q2<...< qw is a sequence of integer scaling factors.

[0101] For each example, families of quantum LDPC codes are considered, wherein the codes Qi, ..., QM in the family are each defined by an 2nmx 2nmparity check matrix Hm, said parity check matrix Hmbeing M of the form wherein 0o, is an q vmL x qmL |mzero-matrix, Amand Bmare qmL x qmL circulant matrices with predetermined binary generating polynomials aqmL(x) = p^xja^x), respectively bqmL(x) = pqm,L(x)bL(x) of degree at most qmL - 1 , wherein pqmiL(x) = 1 is a mutual binary polynomial. aL(x) and bL(x) are predetermined binary generating base code polynomials aL(x) = Xi=oaixi> respectively bL(x) of degree at most L-1 , i.e., the coefficients atand btare either 0 or 1 for i = 1, ..., L - 1, the base code polynomials being such that twice the degree of the greatest common divisor of the base code polynomials aL(x), bL(x) and the binary polynomial (xL-1 ) is equal to a predetermined dimension kv=2deg(gcd(ai(x), bL(x), (xL-1 )))> 2. In order to construct a particular family FLDPC, a suitable base code Qi must be identified. To this end, a search for polynomials aL(x), bi_(x) over the ring is performed. By construction, the generated base code is guaranteed to satisfy the commutativity condition 0. However, it has to be ensured that the polynomials ai_(x), bi(x) satisfy the additional property that deg g(x) > 0, where g(x) = gcd (at(x), bi(x), xL-1 ), otherwise the code has dimension 0 as explained above.

[0102] For the numerical investigations an exhaustive search over all possible pairs of polynomials ai(x), bi_(x) e F2<L>for L up to 10 is carried out. Codes for which deg g(x) = 0 are discarded and it is found that L = 5 is the smallest value that produces error-correcting codes with distance d > 3. To illustrate this, all base codes that satisfy deg g(x) > 0 for L = 4, 5 are presented in Fig. 1. In particular, Fig. 1a depicts the logical error rate (LER) at physical error rate (PER) equal to 0.010. All base codes with non-zero dimension at L = 4 and L = 5 (49 and 226 in total respectively) are included. Codes with L=4 and distance less than three, d<3, are marked with triangles, codes with L=5 and distance less than three, d<3, are marked with crosses and codes with distance at least three, d>3, are marked with squares. It is clear that one needs L > 5 to achieve distance 3, and only codes with distance 3 produce a low LER (below the dashed line at 0.020). One can see that all codes of distance 3 are corresponding to L = 5 and display a low LER, below the dashed threshold line drawn at LER = 0.020, while all codes with distance less than 3 (with L = 4 or 5) produce very high LER, creating a considerable gap from the well-behaving distance 3 codes. Therefore, the smallest codes one can consider have code length 10.

[0103] Since calculation of distance soon becomes costly, the dashed threshold line is extrapolated to higher L. Hence, given a particular L, all codes for which deg g(x) > 0 and LER < 0.020 at PER = 0.010 are considered. Due to an interest in realizing such codes experimentally, the sum of polynomial weights q = wt a(x) + wt b(x) is further limited by 8, so we obtain qLDPC codes with q < 8. In Fig. 1(b), the maximum number of qubits participating in each parity check, equal to the sum of the polynomial weights q = wt (a) + wt (b) is depicted. The codes are ordered in increasing q, and the weight of the considered codes is capped at 8 (dashed line). As L increases beyond 5, this restricts the fraction of considered codes. This is more restrictive at values of L higher than 5, while still allowing for a very large fraction of available codes. Out of those, one gets many examples of codes with L = 5 10 that display improved performance as the code length increases according to the quantum LDPC codes of the present invention. One example of the smallest obtainable family is discussed below, and examples for all L < 10 are presented. The goal is to construct the smallest family TLDPC of LDPC codes, for which all members (except the base code) are extended codes, covered by the present invention. Firstly, L = 5 is selected, the smallest value that guarantees code distance d > 3. Then a search for the base code is performed as described above, which finds the [[10, 2, 3]] base code defined by polynomials ai(x) = 1 + x4and bi(x) = 1 + x + x2+ x4. The parity-check matrix of this code has weight wr= wt a(x) + wt b(x) = 6. It is then proceeded to obtain extended codes from the base code such that the code lengths of the extended codes increase in smallest steps. This is achieved by choosing the integer arithmetic sequence (qi, q2, . . . ) with qm= m for m > 1 , leading to code lengths nm= 10m, i.e. 10, 20, 30, . . . for codes in TTDPC- Finally, we choose PqmL(x) = 1 to ensure a constant wr(m) = 6 for all m, creating the 6-LDPC family TLDPC.

[0104] The performance of the five smallest codes of FLDPC (i.e., the base code and four extended codes) is plotted in Fig. 2. A threshold value at around PER = 0.145 is observed, which is similar to the surface code threshold plotted for comparison. The performance of the first five codes in FLDPC is compared with the surface code of distances d = 3, 5, 7. The performance of codes in ^LDPC improves with increasing code length and the threshold point is around PER = 0.145, similar to the surface code threshold that is plotted. The curve for L=5 depicts the performance of the base code. The quantum LDPC codes of L = 10 and L = 20 are shown to outperform the surface codes of distance 3 and 5 respectively. The quantum LDPC code of L = 25 requires 50 data qubits and 48 syndrome qubits for a total of 98 qubits. In comparison the surface code of distance 7 requires a total of 97 qubits. These two codes appear to exhibit similar performance. Note that the quantum LDPC family encodes 2 logical qubits in contrast to the surface code which always encodes 1 logical qubit.

[0105] Following the same construction, i.e. using a base code with generating polynomials 8L(X), bu(x), qm= m and p(m)(x) = 1 , extended codes which are quantum LDPC code families with performance that improves as the code length increases at every L = 5 10 may be constructed. Examples of these codes are presented below, and their performances are depicted in Fig. 3:

[0106] In the following, extended codes which are quantum LDPC codes of comparable performance are presented. For a given pair of generating base code polynomials ai_(x), bt(x) e F2<L\ the associated base code is constructed and extended according to the method of the present invention explained above using the integer sequence qm= m and pqm,L(x) = 1 for m > 1 . Therefore, extended codes which are quantum LDPC codes with small code length are obtained. Their performances are plotted in Fig. 3 up to m = 4. Each code family is identified by the size Li of its base code. Each plot displays a quantum LDPC code family with given [[2L, ki]] base code L for L1= 5, 6, 7, 8, 9, 10. The base code is defined by the polynomials ai(x), bi(x) e provided below and the base code dimension is given by ki = 2 deg g(x) as explained above, q is the sum of the weights of the base code polynomials, i.e., q = wt 3L(X) + wt bi(x). For each code family, the performance of the smallest four codes (Z1;l2, l3, / 4) is plotted, showing the breakeven point of each code and the threshold point, and compared with the surface code of distances d = 3, 5, 7. The six base codes that are used to create the plots in Fig. 3 are:

[0107] Fig. 3 (a) [[10, 2, 3]] code (L = 5); plotted in Fig. 2, too. ai(x) = 1 + x4; bi(x) = 1 + x + x2+ x4.

[0108] Fig. 3 (b) [[12, 2, 3]] code (L = 6). ai_(x) = 1 + x + x2+ x5; bt(x) = 1 + x + x3+ x5.

[0109] Fig. 3 (c) [[14, 2]] code (L = 7). ai(x) = 1 + x3; bi_(x) = 1 + x + x3+ x6.

[0110] Fig. 3 (d) [[16, 2]] code (L= 8). ai(x) = x + x3; bi(x) = 1 + x5.

[0111] Fig. 3 (e) [[18, 2]] code (L = 9). ai(x) = 1 + x2; bi_(x) = 1 + x5.

[0112] Fig. 3 (f) [[20, 2]] code (L = 10). ai_(x) = 1 + X; bL(x) = 1 + x6.

[0113] Note that all base codes have g(x) = 1 + x, thereby code dimension ki = 2.

[0114] In the plots, the threshold point of each code family along with the breakeven points, i.e. the point at which LER dips below PER for a given code is included. The breakeven points increase as m increases for each code, and the threshold of each family is similar to the surface code threshold. Bigger codes, i.e. higher values of L, allow for a small number of qubits per parity check q = wt a(x) + wt b(x).

[0115] B. Quantum error correction on the basis of extended codes which are members of a family of scalable quantum codes

[0116] In general, part of the description, extended codes Qmwhich are generalized bicycle codes GBfa^bqmi)werepresented, wherein aqmL(x) = pqm,L WaL(x), respectivelybqmiXx) = Pqm,iXx)bL(x) isapolynomial of degree at most qmL - 1 , qm=3m'1, and the mutual binary polynomial is of the form pqm L(x) = The extended codes according to this embodiment may have improved properties with increasing code length, as the km+i> kmfor all base codes, and there are examples for which dm+i > dm.

[0117] In Fig. 4, a chip architecture for implementing the code with m=2 is presented. The chip architecture comprises three identical chips 1 , 2, 3. Each chip comprises a plurality of data and syndrome qubits (black dots) which have the base code connectivity (schematically indicated by the hatching). The base code connectivity is the required connectivity of the base code Qi which is the generalized bicycle code GB(aL, bL) with the generating polynomials aL(x) and bL(x). Further, there is a connectivity 4 between the qubits of the first and the second chip, between the qubits of the second and the third chip, and between the qubits of the first and the third chip such that the connectivity between all qubits is according to the required connectivity of the extended code Ch- In the language used above, the qubits on the first chip may be the first set of qubits, and the qubits on the second and third chip may be the second set of qubits. Further, the qubits on the second chip may be the first module set of qubits and the qubits on the third chip may be the second module set of qubits. It is obvious from Fig. 4 that the scalable codes introduced above are particularly well-suited for implementation on state-of-the-art quantum computing architectures wherein chips with a relatively small number of qubits are replicated and connected to allow for an implementation of QEC codes with larger code length, and potentially larger dimension and distance.

[0118] Fig. 5a - 5j are schematic representations of the required connectivities of the stabilizer operators defined by the X-matrix Hx of the parity check matrix of the base code and extended code for the following example:

[0119] The base code is the generalized bicycle code GB(aL, bL) with generating base code polynomials ai(x) = 1 + x4and bi_(x) = 1 + x + x2+ x4, and L = 5. The extended code is the generalized bicycle code GB(a2L, b2L) wherein the generating polynomials of degree 2L = 10 are of the form a2L(x) = czL(x), b2L(x') = bL(x) (i.e., they are obtained from the base code polynomials by multiplication with the mutual polynomials P2,L(X)=1 ).

[0120] The X-matrix Hx.i of the parity check matrix0of the base code is of the form ffz.l /

[0121] = X, ® X2® X6® X7® ® X10 Si,x,2= X20 X30 X60 X70 X80 X10$1X3 = X30 X40 X60 X70 X80 Xg $i,x,4 = X40 X50 X70 X80 Xg 0 X10$1X5 = Xi 0 X50 X60 X80 Xg 0 X10

[0122] HX 20 \

[0123] The X-matrix Hx,2 of the parity check matrix H2' „ of the extended code is of the

[0124] . 0 nZ 2J form wherein the qubits q; have been relabelled according to qi «-> qi+5 for i=5. 10. The X-matrix of the extended code defines the following 10 stabilizer operators:

[0125] $2X1 = *1 ® *6 ® *12 ® X170 X190 X20$2x2=X20 X60 X70 Xi30 X180 X20$2X3 = X30 X60 X70 X80 X140 X19

[0126] $2X10=X 41 0 X130 X160 X180 X190 X20

[0127] Figs. 5a - 5j each depict an arrangement of 20 data qubits (filled circles, with numbers 1-20) and 10 syndrome qubits (c, i=1. 10, black squares). The syndrome qubit which is used for the implementation of the stabilizer operators indicated in the figures is filled in black, all other syndrome qubits are left with a white filling. The extended code is defined on 20 qubits, and the base code is defined on 10 qubits. In the terminology used in this application, the first set of qubits comprises the data qubits 1 - 10 and the five syndrome qubits Ci,..., C5 arranged between the 10 data qubits. The second set of qubits comprises the data qubits 11 - 20 and the five syndrome qubits c6C10 arranged between said data qubits. For each figure, the connectivities between the qubits in the first set that are only used for the implementation of the stabilizers of the base code are indicated with dashed lines. The connectivities that are used for the implementation of the stabilizer operator of the base code and the extended code are indicated with solid lines. The base code connectivity as defined by each X-stabilizer operator involves one of the syndrome qubits ci c5associated with said stabilizer operator (indicated with filled black square) and at least one of the data qubits 1 - 10 (all solid lines and dashed lines between the one syndrome qubit ci C5 and the plurality of data qubits 1 - 10 in the figures). The second connectivity as defined by each X-stabilizer operators involves only one of the syndrome qubits c6C10 associated with said stabilizer operator (indicated with filled black square) and at least one of the data qubits 11 - 20 (all solid lines between the one syndrome qubit c6C10 and the plurality of data qubits 11 - 20 in the figures). The linking connectivity connects a syndrome qubit of the first (second) set with a data qubit of the second (first) set, see e.g. connection between syndrome qubit Ci and data qubit 12 in Fig. 5a (see e.g. connection between syndrome qubit c& and data qubit 2 in Fig. 5f).

[0128] Obviously, the stabilizer operators S2 Xii, i=1 5 of the extended code require on the first set of data and syndrome qubits only the connectivities of the corresponding stabilizer operators S2:X,i of the base code.

[0129] It is obvious from the structure of the parity check matrix that the second connectivity is also at most the base code connectivity.

[0130] Diagrams similar to the ones shown in Figs. 5a - 5j may also be obtained for the Z- stabilizer operators.

[0131] List of references

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Claims

PATENT CLAIMS1. Method for constructing a quantum error correction code with a code length n’, characterized in that said quantum error correction code is an extended code constructed from a base code, the base code having a code length n being smaller than the code length of the extended code n’, said base code requiring a base code connectivity between qubits of a first set of data and syndrome qubits, wherein said extended code requires a first connectivity between qubits of said first set, a second connectivity between qubits of a second set of data qubits and syndrome qubits, said first and second sets being different from each other, and a linking connectivity for connecting qubits in the two different sets, wherein said first connectivity is at most the base code connectivity.

2. Method according to claim 1 , wherein said extended code is a non-local code.

3. Method according to claim 1 or 2, wherein the extended code further requires a module connectivity between qubits of at least one module set of data and syndrome qubits of said second set, said module connectivity being at most the base code connectivity.

4. Method according to anyone of the preceding claims, wherein the first connectivity and / or the module connectivity is identical to the base code connectivity.

5. Method according to anyone of the preceding claims, wherein the extended code is a CSS code, in particular a quantum Low Density Parity Check code, and in particular the base code is also a CSS code, more in particular a quantum Low Density Parity Check code.

6. Method according to anyone of the preceding claims, wherein the base code is a generalized bicycle code GB(aL, bL) of code length 2L with predetermined binary generating base code polynomials aL(x) = SfjJ ajX1, respectively bL(x) = Xf=o btxlof degree at most L- 1 > 1 , i.e., the coefficients a; and btare either 0 or 1 for i = 0,1, the base code polynomials being such that twice the degree of the greatest common divisor of the base code polynomials aL(x), bL(x) and the binary polynomial (xL-1 ) is equal to a predetermined base code dimension kx=2deg(gcd(aL(x), hL(x), (xL-1 ))) > 2, and the extended code is a generalized bicycle code GB(aqmL, bqmL) of code length 2qmL with predetermined binary generating polynomials aqmL(x) and bqmL(x) of degree at most qmL - 1 , which are related tothe base code polynomials via aqmL(x) = fqm L[aL(x)], respectively bqmL(x) = fqm L[bLW], wherein fqm Lis a function and qm> 1 is an integer scaling factor.

7. Method according to claim 6, wherein the sum of the weights of the base code polynomials aL(x) and bL(x), i.e., the sum of the number of their non-zero coefficients, is at most 10, in particular at most 9, more in particular at most 8, and even more in particular at most 7.

8. Method according to claim 6 or 7, wherein the weight of at least one of the base code polynomials aL(x) and bL(x), i.e., the number of its non-zero coefficients, and in particular the weight of each base code polynomial, is at most 5, in particular at most 4, more in particular at most 3, and even more in particular at most 2.

9. Method according to anyone of claims 6 - 8, wherein the base code polynomials aL(x) and hL(x) and the function fqm Lare such that the sum of the weights of the generating polynomials aqmi(x) and bqmL(x) of the extended code is upper bounded by a constant, and preferably by the sum of the weights of the base code polynomial aL(x) and bL(x).

10. Method according to anyone of claims 6 - 9, wherein either f( / m L[aL(x)]= p(?)njL(x)aL(x) and / (ZmL[bL(x)]= pQm,iW / »L(x)for some mutual polynomial pqm,L(x) or wherein the function fai is a non-linear function.

11. Method according to claim 10, wherein the mutual polynomial is defined as pqmiL(x) =and is a predetermined mutual binary polynomial of degree at most (qm-1 )L, i.e., the coefficients ptare either 0 or 1 for i = 1, ..., (qm- 1)L.

12. Method according to claim 11 , wherein the mutual polynomial is of the form pQm,L(x) = 1 for each m so that the extended code is a quantum Low Density Parity Check code.

13. Method according to claim 10, wherein the scaling factor is qm= 3m’1for an integer m > with the mutual binary14. Method according to anyone of the preceding claims, said method comprising a CSS- code construction step for constructing the base code and the extended code with the following steps: i) selecting the predetermined dimension 2 < / q; ii) selecting two binary test base code polynomials=XfJo b^x1of degree at most L-1 such that twice the degree of the greatest common divisor of the test base code polynomials a®(x), b^(x) and the binary polynomial (xL- 1 ) is equal to the predetermined dimension klti.e., 2deg(gcd(a, (xL-1 )))= ki; iii) constructing a CSS test base code with the base code length ni=2L which is the generalized bicycle code GB (a^, b^) with generating test base code polynomials ap-’(x), b^\x), respectively; iv) selecting a family of test functions f^Lfor a sequence of integer scaling factors 1 < Qi < <?2 < ■" < QM! v) constructing a family of extended CSS test codes with code lengths which are generalized bicycle codes GB (a^L, b^L) , with generating test polynomialsmost qmL- 1 ; vi) selecting a noise model for a system of m, respectively n® qubits, said noise model being described by a quantum channel EAwith an associated noise strength parameter A; vii) simulating, for a plurality of values of the noise strength parameter 2, quantum error correction according the CSS test base code and the extended CSS test codes on a classical computer on a model of m qubits for the CSS test base code, respectively on a model ofqubits for the extended CSS test codes, the qubits being subject to noise according to the noise model with the respective value of the noise strength parameter to thereby obtain for each value of the noise strength parameter an associated logical error rate of the respective code; viii) if for at least one extended test code with scaling factor qmthe logical error rate is below the logical error rate of the CSS test base code, selecting the test base code polynomialsthe predetermined base code polynomials and the CSS code with scaling factor qmas the extended code of the quantum error correction method, otherwise returning to step i).

15. Quantum processing device for implementing a quantum error correction code constructed according to the method of anyone of the preceding claims, said quantum processing device comprises a plurality of data qubits and a plurality of syndrome qubits for implementing the extended code ; characterized in that the quantum processing device comprises the first connectivity between qubits of the first set, the second connectivity between the qubits of the second set, and the linking connectivity for connecting qubits in the two different sets, wherein said first connectivity is at most the base code connectivity.

16. Quantum processing device according to claim 15, wherein the first connectivity is identical to the base code connectivity.

17. Quantum error correction code, said code being the extended code constructed according to anyone of claims 1-14.

18. Method of implementing quantum information using the quantum error correction code constructed according to the method of anyone of claims 1 to 14, in particular for quantum error correction, quantum error mitigation and quantum metrology.

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