A computer-implemented method for quantum compiling for measurement-based

Abstract

A computer-implemented method for quantum compiling for measurement-based unidirectional quantum computation is described. In the method, electronic processing means 2 are provided with digital information corresponding to a unitary operator or matrix U, representing a single-qubit or multi-qubit quantum operation to be carried out, through a measurement-based unidirectional quantum computation, by a quantum computer 1. The unitary operator or matrix are then decomposed in terms of Pauli operators, by the aforesaid electronic processing means 2. The method, then, comprises directly converting, by the electronic processing means 2, the unitary operator or matrix U, decomposed in terms of Pauli operators, into a graph G implementable on the quantum computer 1. The graph G represents a state of a cluster of qubits to be processed by the quantum computer 1 to carry out the desired said quantum operation. In the graph G, each graph vertex represents a respective qubit of the cluster and each edge of the graph corresponds to an entanglement to be imposed between the two qubits corresponding to the two vertices connected by the edge. The aforesaid step of directly converting comprises identifying the sub-set of entanglements to be implemented between qubits of the cluster and identifying the sub-set of qubits of the cluster to be measured, to obtain the graph G representing the state of said cluster, based on the criterion that the measurement of the aforesaid identified sub-set of qubits to be measured affects the output state of the other unmeasured qubits, through the aforesaid identified entanglements, in a manner corresponding to the quantum operation to be carried out. The method finally comprises the step of providing to the quantum computer 1, by the electronic processing means 2, as the result of the quantum compiling, a graph description information Q(G) describing the aforesaid obtained graph (G). A computer implemented method for performing measurement-based unidirectional quantum computation, based on the aforesaid computer-implemented method for quantum compiling, is also described. A quantum computing system, exploiting said computer-implemented methods, is also described.

Description

[0001] “A computer-implemented method for quantum compiling for measurement-based unidirectional quantum computation, and related systems” DESCRIPTION

[0002] TECHNOLOGICAL BACKGROUND

[0003] Field of application.

[0004] The present invention relates to a computer-implemented method for quantum compiling.

[0005] The general technical field of the present invention is quantum computing and quantum compiling for quantum computing.

[0006] More specifically, the invention refers to a computer-implemented method for quantum compiling for measurement-based unidirectional quantum computation, and a related system.

[0007] Description of the Prior Art.

[0008] Quantum computation takes place, at its lowest level, through the transformation of quantum systems, mathematically grounded on unitary matrices acting on the state of n-qubits, where the quantum information is encoded.

[0009] According to the known art, the so called “gate model” is by far the paramount architecture for quantum compiling, adapting physical transformation on a system of qubits to implement logic gates, described as unitary operators.

[0010] However, gate-model quantum computers can in practice perform a limited number of transformations only, due to noise and to constraints in their architecture.

[0011] Therefore, a known alternative to the gate-based model, although less common, is the measurement-based model (i.e., Measurement-Based Quantum Computing - MBQC), which relies on the entanglement of a cluster of qubits and the measurements of a subset of such qubits, affecting the state of the unmeasured output qubits.

[0012] In the light of the above, a strong need is emerging for quantum compiling methods that are applicable to the Measurement-Based Quantum Computing.

[0013] The only known solutions of quantum compiling for Measurement-Based Quantum Computing are based on the so-called “M-Calculus Technique” - see, for example, V. Danos, E. Kashefi, P. Panangaden, “The measurement calculus” (2008); E. Pius, “Automatic parallelization of quantum circuits using the measurement based quantum computing model problem” (2010).

[0014] The M-Calculus technique requires two steps, i.e., a first conversion from the unitary operator U to a quantum circuit described according to the quantum-gate-model, and a second conversion (through M-Calculus) from the quantum-gate-model circuit to a model suitable for measurement-based quantum computing MBQC (also called “one-way quantum computing”). The two-steps conversion required by the known techniques based on M-Calculus is particularly complex and cumbersome.

[0015] Therefore, a need is felt for different, simpler and more effective quantum compiling methods for measurement-based quantum computing.

[0016] The above mentioned technical problem, and the related needs that are particularly felt in the considered technical field, are not solved, up to now, in a satisfactory manner.

[0017] SUMMARY OF THE INVENTION

[0018] It is the object of the present invention to provide a computer-implemented method for quantum compiling for measurement-based unidirectional quantum computation, which allows to solve, at least partially, the drawbacks described above with reference to the prior art and respond to the aforesaid needs felt in the technical field under consideration. Such an object is achieved by a method according to claim 1.

[0019] Further embodiments of such a method are defined in claims 2-12.

[0020] It is a further object of the present invention to provide a computer implemented method for performing measurement-based unidirectional quantum computation, based on the aforesaid computer-implemented method for quantum compiling. This object is achieved by a method according to claim 13.

[0021] It is also an object of the present invention to provide a system capable to carry out the abovementioned methods, and a computer-readable storage medium comprising instructions that allows to carry out the abovementioned methods. This object is achieved by a system according to claim 14 and a computer-readable storage medium according to claim 15.

[0022] Another object of the present invention to provide a quantum computing system, exploiting the above described computer-implemented method. This object is achieved by a system according to claim 16.

[0023] It is also an object of the present invention introduce a new paradigm for quantum compiling, allowing to map any quantum circuit represented by a generic unitary operator acting on a register of qubits, into a cluster state and its corresponding graph.

[0024] The method holds independently from the number of involved qubits and eliminates the transposition from the gate model.

[0025] The theoretical model, associated to the method, provides a system of equations, handling which it is possible to achieve a gauge freedom when converting a unitary operator to a specific graph state.

[0026] The formalism of the theoretical model relies on a group representation of the qubit register, rather than on the state vector or the density matrix representations.

[0027] Exploiting such gauge invariance and group formalism makes feasible to avoid numerical calculus, where the matrix size increases as 2n by the number n of qubits.

[0028] Moreover, the formulation of such system of equations allows to revert already developed graph states to study the gauge invariance beyond a given unitary operator.

[0029] The proposed formulation therefore allows to bypass a processing in a gate-based circuit and compile directly in the MBQC graph format.

[0030] According to an embodiment, this technique is based on the so-called “analysis of the residual”, which allows to reconstruct, starting from the unitary operator decomposed in terms of Pauli matrices, the graph state to be measured and the entanglement between the qubits, corresponding to the vertices of the graph state (while the connections represent entanglement).

[0031] Once the unitary operator is provided in input, the output consists of a graph corresponding to the cluster state.

[0032] The solution of the present invention is based on a direct transposition from the unitary operator to the graph, without having to resort to an intermediate transposition from the gate model.

[0033] BRIEF DESCRIPTION OF THE DRAWINGS.

[0034] Further features and advantages of the method and system according to the invention will become apparent from the following description of preferred embodiments, given by way of indicative, non-limiting examples, with reference to the accompanying drawings, in which:

[0035] - Figure 1 is a simplified block diagram illustrating, in a general way, the difference between the solution of the known art and the solution of the present invention;

[0036] - Figure 2 is a simplified block diagram of a quantum computing system according to an embodiment of the invention, capable of carrying out embodiments of the method according to the invention;

[0037] - Figure 3 and Figure 4 represent respective examples of graphs, used in the method according to the present invention, and some related features.

[0038] DETAILED DESCRIPTION

[0039] A computer-implemented method for quantum compiling for measurement-based unidirectional quantum computation is described, referring to the Figures 1-4.

[0040] The computer-implemented method comprises the step of providing electronic processing means 2 with digital information corresponding to a unitary operator or matrix U, representing a single-qubit or multi-qubit quantum operation to be carried out, through a measurement-based unidirectional quantum computation, by a measurement-based quantum computer 1.

[0041] The method then provides decomposing the unitary operator or matrix U in terms of Pauli operators, by the aforesaid electronic processing means 2.

[0042] Then, the method comprises directly converting, by the electronic processing means 2, the aforesaid unitary operator or matrix U, decomposed in terms of Pauli operators, into a graph G implementable on the quantum computer 1. The graph G represents a state of a cluster of qubits to be processed by the quantum computer 1 to carry out the desired said quantum operation.

[0043] In the graph G, each graph vertex represents a respective qubit of the cluster and each edge of the graph corresponds to the entanglement to be imposed between the two qubits corresponding to the two vertices connected by the edge.

[0044] The aforesaid step of directly converting comprises identifying the sub-set of entanglements to be implemented between qubits of the cluster and identifying the sub-set of qubits of the cluster to be measured, to obtain the graph G representing the state of said cluster, based on the criterion that the measurement of the aforesaid identified sub-set of qubits to be measured affects the output state of the other unmeasured qubits, through the aforesaid identified entanglements, in a manner corresponding to the quantum operation to be carried out.

[0045] The method finally comprises the step of providing to the quantum computer 1 , by the electronic processing means 2, as the result of the quantum compiling, a graph description information Q(G) describing the aforesaid obtained graph G.

[0046] Based on the features described above, the quantum compiling method provides an effective quantum compilation for the Measurement Based Quantum Computing.

[0047] Moreover, due to the flow of the MBQC paradigm (entanglement-measurement), such algorithms can be described both as a register of qubits and as a graph.

[0048] In the latter case, as depicted in the Figure 1 - left side - the qubits are represented as the vertices V of a graph G [V, E], while the entanglements as the edges E of such graph.

[0049] According to an embodiment of the method, the above mentioned step of directly converting provides a direct transposition from the unitary operator or matrix U to the respective graph G, without performing an intermediate transposition from the unitary operator or matrix U to a gate model of the quantum circuit corresponding to the quantum operation to be carried out and a further transposition from the gate model to the graph.

[0050] As illustrated in the Figure 1 , the above-mentioned feature, relating to a direct conversion (i.e., single-step passage) from the unitary matrix to the related graph, clearly distinguishes the method of the present invention from some prior art solutions (already previously mentioned), which require a two-step procedure including a first conversion from the unitary matrix to a gate-model quantum circuit (depicted in the bottom-right corner of Figure 1) and a second conversion (e.g., based on M-Calculus) from the gate-model quantum circuit, thus resulting in a more complex procedure compared to the method of the present invention.

[0051] According to an embodiment of the method, the above mentioned step of directly converting comprises mapping any quantum operation represented by said unitary operator or matrix (U) acting on a register of said qubits into said cluster state and respective corresponding graph G.

[0052] According to an implementation option of the above-mentioned embodiment, the quantum compiling is based on a group representation of said qubit register and on a system of equations based on which a gauge freedom is achieved when converting the unitary operator or matrix U to a specific respective cluster state and respective graph. Such a result is accomplished by writing down the unitary operator U into a linear combination which takes the name of residual. The mapping of the unitary into the cluster state is achieved through a technique which is here called “analysis of the residual”.

[0053] In particular, according to an embodiment of the method, the above-mentioned step of mapping comprises mapping the unitary operator or unitary matrix U into the cluster state by means of a “residual analysis” technique, wherein the residual is associated to a residual operator (R) which is an operator describing the cluster state based on a linear combination.

[0054] According to an embodiment of the method, the unitary operator or matrix is a 2Nx 2Nunitary matrix representing an N-qubit quantum operation.

[0055] According to an implementation option, the step of decomposing the 2Nx 2Nunitary matrix U in terms of Pauli operators is based on a KAK decomposition.

[0056] According to other possible implementation options, the step of decomposing the 2Nx 2Nunitary matrix U in terms of Pauli operators is based on any decomposition in set of terms of universal gates (for example, H, T, CX).

[0057] According to an embodiment of the method, the aforesaid step of directly converting is carried out based on the following theorems:

[0058] [TH1]: if a measured i-th qubit was entangled with a j-th output qubit, the i-th active coefficient (Bi) in a residual operator (R) will display a Z quantum operator (Zj) of the j-th output qubit, wherein the residual operator (R) is an operator describing the cluster state after having performed the measurements on the qubits to be measured;

[0059] [TH2]: if two measured qubits were neighbouring qubits, the product of their active coefficients will display a minus sign in front of them.

[0060] According to a specific implementation option, the aforesaid step of directly converting comprises the following sub-steps:

[0061] - describing the quantum state | ’> of n non-entangled qubits by means of the following tensor product (1): where a, and bi are, respectively, the idle and active coefficients of the generator operator G of the i-th non-entangled qubit;

[0062] - describing the entanglement of each pair of two qubits (i,j) to be entangled by applying the corresponding controlled-phase quantum operator CZij thus obtaining an operator Ki according to the relationship: so that the equation (1) is mapped into a respective cluster state;

[0063] - describing the measurement of each qubit to be measured based on the following equation (2): ft) where and Pi are, respectively, the idle and active coefficients of the measurement;

[0064] - obtaining said residual operator (R), describing the cluster state after having performed the measurements on the qubits to be measured, according to the following formula: where the unmeasured qubits run from m+1 to n+m;

[0065] - expressing the residual operator (R) based on the residual idle coefficients (Ai) and active coefficients (Bi), wherein said residual idle coefficients (Ai) and active coefficients (Bi) are obtained from the generator idle coefficients (ai) and active coefficients (bi) and from the measurement idle coefficients (ai) and active coefficients (Pi) according to the equations: <4 .4 ; / r i Bi

[0066] In accordance with an embodiment of the method, the unitary operator is a single-qubit operator.

[0067] In this case, the step of directly converting is based on the following equations: where: q is the unitary supplement operator .16 + Bb Aa - Ba, where A, B are the residual coefficients, a, b are the generator coefficients;

[0068] T is the transfer tensor, corresponding the unitary supplement operation rewritten in Pauli basis (I, X, Z, ZX) f ■la Ab

[0069] Aal 4- Ah 4- BaZ + BbZX Ba Bb

[0070] M is an isomorphism between the two basis, i.e., M maps q to T, and M-1 T to q:

[0071] S is a gauge operator, describing gauge freedom conditions.

[0072] According to an implementation example, the aforesaid gauge operator is the following fully symmetrized gauge operator:

[0073] According to an embodiment of the method, the unitary operator is a multi-qubit operator.

[0074] In this case, the step of directly converting is based on the following equations: where q is the unitary supplement operator, T is the transfer tensor, M is an isomorphism between the two basis, where Mjj = Miiji Mi2j2 ... Minjn and i ° j stands for all the possible compositions between the components of the two vectors, and Sij is a gauge tensor.

[0075] According to an implementation example, the aforesaid gauge tensor Sij , for a register of n output qubits, can be described as S®S®...®S repeated n times.

[0076] In this case, the resulting transfer tensor is: wherein, after mapping the 1 / 2npre-factor into 2n+1coefficients of value 1A / 2, the To matrix elements shape into and the TOj elements read

[0077] A computer implemented method for performing measurement-based unidirectional quantum computation is here described.

[0078] This method comprises the steps of determining by a quantum computer 1 a singlequbit or multi-qubit quantum operation to be carried out, represented by a respective unitary matrix; then, providing said unitary matrix U to electronic processing means 2.

[0079] The method then provides carrying out a computer-implemented method for quantum compiling according to any one of the previously described embodiments.

[0080] The method finally comprises the steps of implementing the graph G provided by the method of quantum compiling, and executing the quantum operation, by the quantum computer 1 , by measuring the sub-set of qubits to be measured and reading the quantum operation results on the output qubits of the graph.

[0081] A system is here described, comprising one or more computers and one or more storage devices storing instructions that are operable, when executed by the one or more computers, to cause the one or more computers to perform operations comprising the methods of any one of the previously described embodiments.

[0082] There is here described a computer-readable storage medium comprising instructions stored thereon that are executable by a processing device and upon such execution cause the processing device to perform operations comprising the method of any one of the previously described embodiments.

[0083] A measurement-based quantum computing system, also comprised in the invention and exploiting the above described computer-implemented method for quantum compiling, is hereafter described, referring in particular to Figure 2.

[0084] The measurement-based quantum computing system includes a quantum computer 1 , comprising a control unit 11 and a quantum computation unit 10, and quantum compiler electronic processing means 2 (e.g., a quantum compiler electronic processor 2 - above also indicated as “electronic processing means 2”).

[0085] The control unit 11 of the quantum computer 1 is configured to generate a unitary matrix or operator U, representing a single-qubit or multi-qubit quantum operation to be carried out by the quantum computation unit 10, and is further configured to act on the quantum computation unit 10 to implement a graph G, representing the quantum operation to be carried out, based on a received graph description information Q(G) describing the graph G in terms of vertices and edges, wherein each graph vertex represents a respective qubit of the cluster and each edge of the graph corresponds to an entanglement to be imposed between the two qubits corresponding to the two vertices connected by the edge.

[0086] The quantum computation unit 10 is configured to implement a quantum circuit corresponding to said graph G, under the control of the control unit 11 , and to perform the single-qubit or multi-qubit quantum operation by means of the quantum circuit.

[0087] The quantum compiler electronic processor 2 is configured to receive the unitary matrix U from the control unit 11 of the quantum computer 1 , and to carry out a computer-implemented method for quantum compiling, according to any of the method embodiments described above.

[0088] A theoretical description of the approach adopted for this invention is reported here below, for illustrating purposes, describing examples not limiting the scope of the invention.

[0089] In the first place, we describe a system of non-entangled qubits via the so-called generators G:

[0090] A multi-qubit register of non-entangled qubits can be thus described via a tensor product:

[0091] In such a formalism, it is possible to entangle two qubits via the controlled-phase operators CZ thanks to the following property:

[0092] Therefore, the generators G are reshaped in the following way:

[0093] 4 X1XiZi = K,

[0094] In such a way, it is possible to map the state in Equation (1) into a cluster state.

[0095] The next step consists of embedding the quantum measurements in such formalism: a measurement over a qubit is a projection over a particular subspace of a Hilbert space C2 (i.e. , a 2D vector space over the complex field), which can be represented as:

[0096] After having performed several measurements over the cluster state, we are left with the generators of the unmeasured qubits and a linear combination of Z operators, carrying the indices of such output qubits, which we call to be the “residual” operator R: The unmeasured qubits, in the above Equation, run from m+1 to n+m.

[0097] As a practical example, in Figure 3 a graph with 5 qubits is depicted, wherein the qubits indicated as “4” and “5” are the ones to be measured. The dashed line means that the entanglement between the qubits “4” and “5” can hold or not, just to show how the residual will be affected by such variation.

[0098] A variable is introduced; if the entanglement holds, s=1 , otherwise s=0.

[0099] The cluster state can be described as follows:

[0100] The “a” are defined to be the idle coefficients, as no Pauli operator is carried by them. Instead, the “bi” are defined as the active coefficients.

[0101] After the quantum measurements, the residual assumes the following form: where the Ai, Bi coefficients are respectively the idle and active coefficients of the residual R, and derive from both the active and idle coefficients of the measurements and the generators in the following way: i.e., the product between the coefficients from both the generators and the measurements in Equation (1) and (2) respectively.

[0102] The paramount achievement of the above reported theoretical description can be summed up by the two theorems [TH1] and [TH2] already mentioned above.

[0103] A more detailed description of a method embodiment, applicable to single qubit operators, is reported here below.

[0104] When the output qubit is a single qubit, it is possible to condense the residual and the single-qubit generator in the following form (3):

[0105] The linear combination of operators can indeed be mapped into an operator, which we call the unitary supplement q: The same operator can be rewritten in the Pauli basis (I, X, Z, ZX) instead of the canonical one: we called this operator, rewritten in such a new basis, the transfer tensor T:

[0106] Nevertheless, in order to describe the isomorphism M between these two basis, a vector notation could turn to be useful: where M maps q to T, and M-1 T to q.

[0107] In Equations (4) and (5), T and q are set in what can be called the “matrix format”, i.e. , they are represented as matrix, despite of the number of qubits we are dealing with.

[0108] In Equation (6), instead, the two tensors are represented to be 1-rank, i.e., tensors carrying a single index. While the matrix format maintains the formulation of q and T as operators acting over the |0> state, the tensor format helps manipulating the same tensors via linear transformations.

[0109] In order to match the linear combination in (3), the T tensor must obey to the following symmetry: which condition can be summed up by the following Equation: or with a subtler formalism: where the second Equation is nothing but the first one, with the group structure induced by the following Table:

[0110] With reference to the “gauge freedom”, it is noted that, in order to tackle an MBQC formulation of the problem, the symmetry Equation (7) must be accomplished.

[0111] Once the unitary supplement is provided, the sole condition to make the original transformation work is the following: i.e., the sole first column of q matters in order to make the overall algorithm work.

[0112] Such condition can be stated as (8): which means that q can be transformed as desired, as long as the first column preserves the same form.

[0113] In order to visualize the symmetries over T to be respected, a new operator S can therefore be introduced, which is here called the “gauge operator”:

[0114] (9)

[0115] The above-mentioned conditions (6), (7), (8), (9) can eventually be expressed by the following system of Equations:

[0116] A practical example of gauge transformation that is adopted in this embodiment, scalable with any number of qubits, is the gauge here called “fully symmetrized”, because it is induced by the following matrix for the case 1 -qubit:

[0117] A description is reported here below with the purpose to extend the formalism from a number of qubits n=1 to a number of qubits n=2, with the aim to generalize for any n.

[0118] In the first place, the residual and the generators of the system can be described by the unitary supplement as: from which it derives the transfer tensor T, in both the matrix and tensor format respectively:

[0119] The way the coefficients are disposed in the above matrices, with respect to their corresponding generators, can be described by the following two Tables (10):

[0120] Matrix format Tensor format

[0121] For n=2, the transfer tensor T in the tensor format carries two indices, therefore the linear transformation on it are induced as follows:

[0122] Whereas the M isomorphisms act each over each index, the S gauge transformation is not constrained to be factorized into two terms Siii2 Sjij2. With a more compact notation, the same Equation can be rewritten as follows (Equation 11): The condition (8) still holds for q in the matrix format. Indeed, the condition (7) should be applied on either or both the iiji indices: where (h , ji) ° 02, j?) can results into one of the following four outcomes: (h , ji), (h ° i2, ji), (ii , ji ° j?), (ii ° is, ji ° js), while (i2, j?) ° (ii , ji) into (i2, js), (i? ° h , j?), (i?, j?0ji), O20ii , j20ji).

[0123] All the previous compositions must be accomplished in Equation (11) by the T coefficients.

[0124] According to an embodiment, the method can be applied to multi-qubits operator, as described here below.

[0125] The cardinal equations for MBQC gauge compiling, valid also for the multi-qubit operator case, are:

[0126] T 1 ;’T“ / “J"? I j Jj ™ J ioj-ljoi where Mij = Miiji Mi2j2 ... Minjn and i ° j stands for all the possible compositions between the components of the two vectors.

[0127] Indeed, the Sy tensor can assume any form till the gauge equation is respected.

[0128] The fully symmetrized gauge Sij, for a register of n output qubits, can be described as S®S®...®S repeated n times.

[0129] Once such operator is applied to the transfer tensor, the form of the new T is provided by

[0130] The 1 / 2npre-factor can be mapped into 2n+1coefficients of value 1 A / 2.

[0131] Each T term is therefore multiplied as follows: the To matrix elements shape into: where the Pauli matrices follow the disposition of the Table in the matrix format from Equation (10).

[0132] On the other hand, the Toj elements read:

[0133] While the ai, bi coefficients stand for the output qubits, the Ai, Bi ones represent the qubits of the residual connected to the outputs, while the coefficients from all the other qubits are absorbed into the Ty terms.

[0134] An illustration is provided in the Figure 4: all the unnamed qubits, in the left part of the depicted graph, derive from Ty, i.e., from the transfer tensor before the gauge transformations; instead, the qubits indicated as Bi ... Bnare added by the fully symmetrized gauge transformation; finally, the qubits indicated as Oi ... On, in the right part of the depicted graph, are the outputs of the computation.

[0135] An example of a quantum operation, which can be carried out by the method of the present invention, is reported here below, for illustrative and non-limiting purpose.

[0136] The considered quantum operation is a rotation around the X-axis.

[0137] The action of the Rx(0) operator, in terms of unitary supplement, can be depicted as:

[0138] In the canonical basis, such operation assumes the following matrix form: For sake of simplicity, we now rename C=cos(0 / 2) and S=sin(0 / 2). The transfer matrix reads as which does not match however the required symmetries in the cardinal Equations (13).

[0139] The action by the fully symmetric gauge yields:

[0140] Applying the rules from the above reported Equations (14) and (15), the transfer tensor morphs into

[0141] The A and B elements are, respectively, the idle and the interactive coefficients cast into the residual, while Opand Oi are the idle and interactive coefficients for the output qubit.

[0142] All these new coefficients (A, B, Opand Oi) are set to 1A / 2.

[0143] We are interested in the sole Too and T21 terms, the first one representing the coefficient associated to the I basis element, the second one with the Z.

[0144] As no minus sign occurs between the interacting terms b and iS, the respective qubits are not neighbours.

[0145] On the contrary, a minus sign appears in front of the Cb and iSa terms, suggesting that such qubits neighbour with the G(A,B) one.

[0146] For construction, the G(A,B) and the G(Op,Oi) interact among them.

[0147] Within such scheme, the measurements over all the qubits are given by Oi [1 -I - J / x / 2 i.e., by a projection over the |+> state. Nevertheless, it is possible to sum over C+iS =eie / 2:

[0148] It is possible to trace out an irrelevant global phase e'9, making the transfer tensor equal to:

[0149] Applying the exposed Theorem, it is possible to split e-'9and B as two interactive coefficients, from a measurement and from a generator respectively, b interacting with B and therefore yielding the minus sign.

[0150] The corresponding algebraic description, in terms of generators, is provided by

[0151] [«. + bXlZ2][A + BZ,X2Z3][OP+ 0^2X3]

[0152] Recalling that A=B=Oi=Op=1 A / 2, the overall result is:

[0153] — [ft

[0154] This equation expresses the stabilizer state to be measured, in order to achieve the rotation around the x-axis via the fully symmetrized gauge.

[0155] The first qubit, pinned by Xi, is measured in the |+> basis, while the second one, provided by the Z1X2Z3 generator, is projected into the following state:

[0156] It should be noticed that the object of the present invention is fully achieved by the method described above by virtue of its functional features, as described above in detail.

[0157] In order to meet contingent needs, those skilled in the art can make changes and adaptations to the embodiments of the method and system described above, and can replace elements with others which are functionally equivalent, without departing from the scope of the following claims. All the features described above as belonging to a possible embodiment may be implemented irrespective of the other embodiments described.

Claims

1. CLAIMS1. A computer-implemented method for quantum compiling for measurement-based unidirectional quantum computation, comprising:- providing electronic processing means (2) with digital information corresponding to a unitary operator or matrix (U), implementing a single-qubit or multi-qubit quantum operation to be carried out by a measurement-based quantum computer (1);- decomposing said unitary operator or matrix (U) in terms of Pauli operators, by said electronic processing means (2);- directly converting, by the electronic processing means (2), said unitary operator or matrix (U) decomposed in terms of Pauli operators into a graph (G) implementable on said quantum computer (1), said graph (G) representing a cluster state of qubits to be processed by the quantum computer to carry out said quantum operation, wherein each graph vertex represents a respective qubit of the cluster state and each edge of the graph corresponds to the entanglement to be imposed between the two qubits corresponding to the two vertices connected by the edge, wherein said step of directly converting comprises identifying the sub-set of entanglements to be implemented between qubits of the cluster state and identifying the subset of qubits of the cluster state to be measured, to obtain said graph representing said cluster state, based on the criterion that the measurement of said identified sub-set of qubits to be measured affects the output state of the other unmeasured qubits, through said identified entanglements, in a manner corresponding to the quantum operation to be carried out;- providing to said quantum computer (1), by said electronic processing means (2), as the result of the quantum compiling, a graph description information (Q(G)) describing said obtained graph (G).

2. Method according to claim 1 , wherein said step of directly converting provides a direct transposition from the unitary operator or matrix (U) to the respective graph, without performing an intermediate transposition from the unitary operator or matrix (U) to a gate model of the quantum circuit corresponding to the quantum operation to be carried out and a further transposition from the gate model to the graph.

3. Method according to claim 1 or claim 2, wherein said step of directly converting comprises mapping any quantum operation represented by said unitary operator or matrix (U)acting on a register of said qubits into said cluster state and respective corresponding graph.

4. Method according to claim 3, wherein the quantum compiling is based on a group representation of said qubit register and on a system of equations based on which a gauge freedom is achieved when converting the unitary operator or matrix (U) to a specific respective cluster state and respective graph.

5. Method according to claim 3 or claim 4, wherein said step of mapping comprises mapping the unitary operator or unitary matrix U into the cluster state by means of a “residual analysis” technique, wherein the residual is associated to a residual operator (R) which is an operator describing the cluster state based on a linear combination.

6. Method according to any of the preceding claims, wherein:- the unitary operator or matrix is a 2Nx 2Nunitary matrix representing an N-qubit quantum operation;- said step of decomposing the 2Nx 2Nunitary matrix (U) in terms of Pauli operators is based on a KAK decomposition, or on any decomposition in set of terms of universal gates.

7. Method according to any of the preceding claims, wherein said step of directly converting is carried out based on the following theorems:- [TH1] if a measured i-th qubit was entangled with a j-th output qubit, the i-th active coefficient (Bi) in a residual operator (R) will display a Z quantum operator (Zj) of the j-th output qubit, wherein the residual operator (R) is an operator describing the cluster state after having performed the measurements on the qubits to be measured;- [TH2] if two measured qubits were neighbouring qubits, the product of their active coefficients will display a minus sign in front of them.

8. Method according to claim 7, wherein said step of directly converting comprises:- describing the quantum state | ’> of n non-entangled qubits by means of the following tensor product:where a, and bi are, respectively, the idle and active coefficients of the generator operator G of the i-th non-entangled qubit;- describing the entanglement of each pair of two qubits (i,j) to be entangled by applying the corresponding controlled-phase quantum operator CZij thus obtaining an operator Ki according to the relationship:so that the equation (1) is mapped into a respective cluster state;- describing the measurement of each qubit to be measured based on the following equation (2)where and Pi are, respectively, the idle and active coefficients of the measurement;- obtaining said residual operator (R), describing the cluster state after having performed the measurements on the qubits to be measured, according to the following formula:where the unmeasured qubits run from m+1 to n+m;- expressing the residual operator (R) based on the residual idle coefficients (Ai) and active coefficients (Bi), wherein said residual idle coefficients (Ai) and active coefficients (Bi) are obtained from the generator idle coefficients (ai) and active coefficients (bi) and from the measurement idle coefficients (cQ and active coefficients (Pi) according to the equations:

9. Method according to any of the preceding claims, wherein the unitary operator is a single-qubit operator, and wherein said step of directly converting is based on the following equations:where: q is the unitary supplement operatorwhere A, B are the residual coefficients, a, b are the generator coefficients;T is the transfer tensor, corresponding the unitary supplement operation rewritten in Pauli basis (I, X, Z, ZX):M is an isomorphism between the two basis, i.e., M maps q to T, and M-1 T to q:S is a gauge operator, describing gauge freedom conditions.

10. Method according to claim 9, wherein said gauge operator is the following fully symmetrized gauge operator:

11. Method according to any of the claims 1-8, wherein the unitary operator is a multi-qubit operator, and wherein said step of directly converting is based on the following equations:where q is the unitary supplement operator, T is the transfer tensor, M is an isomorphismbetween the two basis, where Mjj = Miiji Mi2j2 ... Minjn and i ° j stands for all the possible compositions between the components of the two vectors, and Sy is a gauge tensor.

12. Method according to claim 11, wherein the gauge tensor Sij , for a register of n output qubits, can be described as S®S®...®S repeated n times, and wherein the resulting transfer tensor is:wherein, after mapping the 1 / 2npre-factor into 2n+1coefficients of value 1A / 2, the To matrix elements shape intoand the TOj elements read13. A computer implemented method for performing measurement-based unidirectional quantum computation comprising:- determining by a quantum computer (1 ) a single-qubit or multi-qubit quantum operation to be carried out, represented by a respective unitary matrix or operator (U);- providing said unitary matrix (U) to electronic processing means (2);- carrying out a computer-implemented method for quantum compiling according to anyof the claims 1-12;- implementing the graph (G) provided by the method of quantum compiling, and executing the quantum operation, by the quantum computer (1), by measuring the sub-set of qubits to be measured and reading the quantum operation results on the output qubits of the graph.

14. A system comprising one or more computers and one or more storage devices storing instructions that are operable, when executed by the one or more computers, to cause the one or more computers to perform operations comprising the methods of any one of claims 1 to 12.

15. A computer-readable storage medium comprising instructions stored thereon that are executable by a processing device and upon such execution cause the processing device to perform operations comprising the method of any one of claims 1 to 13.

16. A measurement-based quantum computing system comprising:- a measurement-based quantum computer (1 ) comprising a control unit (11) and a quantum computation unit (10), wherein the control unit (11 ) is configured to generate a unitary matrix or operator (U), representing a single-qubit or multi-qubit quantum operation to be carried out by the quantum computation unit (10), and is further configured to act on the quantum computation unit (10) to implement a graph (G), representing the quantum operation to be carried out, based on a received graph description information (Q(G)) describing the graph (G) in terms of vertices and edges, wherein each graph vertex represents a respective qubit of the cluster and each edge of the graph corresponds to an entanglement to be imposed between the two qubits corresponding to the two vertices connected by the edge, and wherein the quantum computation unit (10) is configured to implement a quantum circuit corresponding to said graph (G), under the control of the control unit (11 ), and to carry out quantum measurements on the sub-set of qubits to be measured, to perform said singlequbit or multi-qubit quantum operation by means of said quantum circuit;- quantum compiler electronic processing means (2) configured to receive said unitary matrix (U) from the control unit (11) of the quantum computer, and to carry out a computer- implemented method for quantum compiling, according to any of the claims 1-12.

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