Method of quantum error reduction, quantum computer, classical computer, and system

Abstract

A method of quantum error reduction applied to a quantum circuit to be executed on a quantum processing unit, wherein the quantum circuit has one or more symmetries, each represented by a symmetry operator, and the quantum processing unit is provided with a noise characterization, each operator element of the noise characterization represented by a noise operator, the method comprising the steps of: identifying at least noise operators that anticommute with at least one symmetry operator as detectable errors; identifying noise operators not comprised in the detectable errors as non- detectable errors; performing a first error reduction process of reducing the non-detectable errors; and performing a second error reduction process of reducing the detectable errors.

Description

[0001] Method of quantum error reduction , quantum computer , classical computer , and system

[0002] Technical field

[0003] The present disclosure relates to a method of quantum error reduction applied to a quantum circuit to be executed on a quantum processing unit . It moreover relates to a quantum computer having a quantum processing unit configured to implement the method of quantum error reduction, a classical computer having a classical processing unit configured to implement the method of quantum error reduction and a system of the quantum computer and the classical computer .

[0004] Background

[0005] In the technical field of quantum computing, one of the more prominent , i f not the most prominent model , is the quantum circuit ( QC ) model . This model can be seen as a quantum version of the classical circuit model used in classical computing in which classical bits are replaced by qubits ( short for quantum bits ) and quantum operations replace classical operations .

[0006] In a QC, the quantum operations may be represented by quantum gates . In general , a quantum gate ( or simply gate ) refers to a fundamental physical operation that manipulates the quantum state of one or more qubits . The skilled person understands that quantum gates may be represented by matrices and qubits ( or more precisely : the quantum state of the qubit ) may be represented by vectors or matrices, and the application to qubits involves matrix multiplications. When a quantum gate is applied to one or more qubits, it transforms the quantum state of these one or more qubits. Performing a gate on one or more qubits may thus be considered as involving a manipulation of the corresponding quantum state (s) of the qubit (s) according to the specific operations defined by the gate.

[0007] However, quantum devices, in particular in the current era of Noisy Intermediate-Scale Quantum (NISQ) devices, are often subject to substantial noise leading to faulty quantum operations. To tackle these faulty quantum operations, various techniques of quantum error reduction have been developed. The two main means of quantum error reduction are quantum error correction, where a faulty computation is analyzed and the error is corrected, i.e., removed, and quantum error mitigation in which the error is not outright removed but instead mitigated, for example by counter-acting against sources of error. In other words, quantum error mitigation can be understood as improving the accuracy of the computations at the cost of more runs of the quantum computer.

[0008] While quantum error correction, for example by means of so- called error correction codes, is possible, at the present stage of development, the technical requirements for it to be feasible are not met yet. This is in particular due to the nocloning theorem of quantum mechanics which requires quantum error correction to use more advanced techniques than usable in classical computation.

[0009] Examples for such quantum error mitigation techniques are probabilistic error cancellation (PEC) , see also E. van den Berg et al. : "Probabilistic error cancellation with sparse Pauli-Lindblad models on noisy quantum processors", Nature Physics volume 19, pages 1116-1121 (2023) , in which the noise is counteracted by adding gates to the quantum circuit, thus probabilistically cancelling the errors; zero noise extrapolation ( ZNE ) , in which the noise is ampli fied such that based on results with di f ferent levels of noise a result with zero noise can be extrapolated; and post selection ( PS ) for simulations of fermionic systems in which it is exploited that an error- free simulation of a fermionic system conserves the number of fermions is the system, see also R . Chien et al . , https : / / arxiv . org / abs / 2303 . 02270 .

[0010] It should however be born in mind that even tough techniques for quantum error mitigation have been developed, also these techniques scale exponentially both in the number of qubits and the amount of noise in the system .

[0011] Moreover, present techniques are either completely agnostic to the underlying quantum circuit and thus do not exploit the possibility to improve the quantum error mitigation, in particular the scaling of the required number of additional runs of the quantum computer with the number of qubits and the amount of noise , in view of the details of the speci fic quantum circuit or, i f they are based on exploiting symmetries in the underlying quantum circuit , are limited to a certain set of errors related to these symmetries and thus , while potentially showing a better scaling in the number of qubits and the amount of noise , are limited in their capability to mitigate errors .

[0012] In view of these existing approaches , there is a need for quantum error reduction techniques that combine the exploitation of the quantum circuit to which they are applied to such that the resources necessary are be reduced while at the same time ensuring that all errors potentially occurring in the quantum processing unit can be addressed .

[0013] Summary

[0014] The present disclosure has been made in view of the above technical limitations of currently existing quantum error reduction methods and thus provides a quantum error reduction method that provides a more ef f icient use of the provided resources , that is , achieves a similar level of quantum error reduction with less resources or achieves an improved level of quantum error reduction with the same resources when compared to conventional techniques .

[0015] According to an aspect of the present disclosure , a method of quantum error reduction applied to a quantum circuit to be executed on a quantum processing unit is provided, wherein the quantum circuit has one or more symmetries , each represented by a symmetry operator, and the quantum processing unit is provided with a noise characteri zation, each operator element of the noise characteri zation represented by a noise operator , the method comprising the steps of : identi fying noise operators that anticommute with at least one symmetry operator as detectable errors ; identi fying noise operators not comprised in the detectable errors as non-detectable errors ; performing a first error reduction process of reducing the non-detectable errors ; and performing a second error reduction process o f reducing the detectable errors .

[0016] According to another aspect of the present invention, a quantum computer having a quantum processing unit is provided, wherein the quantum computer is configured to implement the above method of determining a noise characteri zation .

[0017] According to a further aspect o f the present invention, a classical computer having a classical processing unit i s provided, wherein the classical computer is configured to implement the above method of determining a noise characteri zation .

[0018] According to a further aspect o f the present invention, a system of the above quantum computer and the above classical computer is provided . In addition, preferred aspects of the present disclosure are defined in the dependent claims .

[0019] According to the invention, a more ef ficient quantum error reduction method can be provided .

[0020] Brief description of the drawings

[0021] Embodiments of the present disclosure , which are presented for better understanding the inventive concepts , but which are not to be seen as limiting the disclosure , will be described with reference to the figures in which :

[0022] Fig . 1 shows a flow chart for a method of quantum error reduction;

[0023] Fig . 2 shows an illustration of detectable and non- detectable errors ;

[0024] Fig . 3 shows a comparison of the relative cost ( left ) and the error ( right ) of methods according to the present invention with conventional approaches of quantum error reduction; and

[0025] Fig . 4 shows comparisons of the mean-square-error of methods according to the present invention with conventional approaches of quantum error reduction .

[0026] Detailed description

[0027] As the present disclosure relates to the technical field of quantum computing, in particular to details of how to characteri ze the noise present when executing a quantum gate / a quantum circuit , the following paragraphs will provide further details regarding the technology referred to and the terms used within this disclosure to facilitate the understanding o f the present disclosure and the inventive concepts disclosed herein .

[0028] Quantum computing can generally be understood as technically implemented computing based on or exploiting quantum mechanical phenomena . Under certain conditions , in particular at small scales , classical theories of physical matter have to be replaced by quantum theories . One core element of these theories is that physical matter exhibits properties of both particles and waves . Quantum computing is built on the fact that leveraging this behavior can lead to what is a called a "quantum advantage" : For some calculations , there exist quantum algorithms that outperform classical algorithms , i . e . , algorithms performed on a classical computer, by a substantial margin, in some cases even exponentially faster .

[0029] However, quantum computing as understood at present may not replace classical computing in general and for every type o f calculation but only for speci fic technical applications for which a quantum algorithm outperforming known classical algorithms is known . Typical examples thereof include Shor' s algorithm for finding prime factors of an integer showing an exponential speedup compared to known classical algorithms and Grover' s algorithm for an unstructured search showing a quadratic speedup compared to known classical algorithms , both having a wide range of possible applications . Further fields where quantum computing is expected to outperform classical computing is the field of quantum simulation, i . e . , simulating a quantum system by using another quantum system governed by equivalent equations , originally proposed by Richard Feynman as well as speci fic optimi zation problems , in particular hybrid algorithms combining quantum computing aspects with classical optimi zation techniques . These two examples are followed by various industries as they could improve the performance and feasibility of many computationally very demanding tasks such as drug discovery and drug development , logistics as well as engineering .

[0030] At the same time , the fragile nature of quantum states leads to the possible computational advantage from quantum computing to be closely tied to a demanding engineering challenge as the quantum behavior of these states has to be preserved for a suf ficient amount of time . Due to the presence of noise disrupting the quantum behavior, the number of operations that can be performed on a quantum computer are limited and as a consequence , large-scale algorithms cannot be reali zed on the currently available Noisy Intermediate-Scale Quantum (NISQ) devices , i . e . , devices with non-negligible noise and for which scaling the quantum computer, that is , increasing the number of qubits while maintaining a high level of qubit quality across all qubits , remains a challenge .

[0031] In this context , reducing the errors due to noise by mitigating the errors has been proven to be a valid strategy to improve the functionality and practicability of currently available NISQ devices and thus further improvements in this area are understood to be a cornerstone for application-oriented quantum computing before suf ficiently good system performance is achieved so that large-scale error correction is possible .

[0032] As quantum computing originates in quantum physics , but relates to the field of computer technology, there is a need for a model or representation to bring quantum physics and computer technology together . The model currently most established i s the QC model based on the classical circuit model . In the ( classical ) circuit model , a ( classical ) circuit is comprised of bits , having either the value 0 or the value 1 , to which gates are applied . In the QC model , each of these elements is replaced by its "quantum version" .

[0033] The quantum version of the bit is the qubit ( also referred to as quantum bit ) . It is , similar to a classical bit , a two- level (or two-state) system, however, a quantum-mechanical two-level system, possibly an effective two-level system. As a consequence of quantum physics, a qubit may be in any coherent superposition of both states 0 and 1 simultaneously.

[0034] A qubit or quantum bit may be considered as the basic unit of quantum information technology as well as the two-level quantum-mechanical system. It may refer to a physical qubit (that is, physically implemented qubit) and / or a logical qubit. As the present disclosure relates to quantum error reduction, the focus will be on the physical qubit.

[0035] A quantum gate (or simply gate) may be considered as the basic quantum circuit operating on one or more qubits. Depending on whether one refers to the logical qubits or the physical qubits, the quantum gate may thus either refer to an operation on logical qubits, or to an operation in the context of the physical quantum-mechanical two-level system, i.e., a quantum gate operating on the physical qubits (as implemented by hardware, the physical quantum system) .

[0036] Quantum gates may operate on a various number of qubits. If it operates only on one qubit, the gate is also called a "singlequbit gate". Accordingly, "two-qubit gates" operate on two qubits. While also gates operating on three or more qubits are possible, for most applications, only single- and two-qubit gates are used.

[0037] A quantum circuit may be considered as a set of quantum gates, in particular a set of quantum gates spanning one or more layers .

[0038] A layer of quantum gates, as part of the quantum circuit, may be considered as (comprised of) a set of quantum gates of the quantum circuit that can be executed between two points in time. Here, executing gates may mean that the on the physical quantum systems operations are carried out that correspond to the execution of the quantum gates on the physical qubits .

[0039] A noise characteri zation of a quantum processing unit may be considered as the characteri zation of the properties of the quantum processing unit related to the noise that the quantum processing unit is subj ected to . While the noise characteri zation may be considered to mainly relate to the quantum processing unit , it cannot be considered separately from the gates to be executed . Moreover, while such a noise characteri zation may mostly depend on the individual ( two qubit ) gates , it may also depend on potential ( spatial ) cross talk and other phenomena . That is , the noise characteri zation may not necessary depend only on the choice of qubits and operations but may depend on the whole quantum circuit and the whole quantum processing unit .

[0040] At present , there are various possible platforms for quantum computing . One of the more prominent ones are superconducting circuits featuring one or more non-linear Josephson j unctions used as the physical system of the (physical ) qubits .

[0041] Moreover, even though the manufacture of such superconducting circuits is done by high precision machines , at the scales of quantum computing, two quantum processing units are rarely i f ever equal and miniscule di f ferences in manufacture but also within the quantum processing unit or the quantum computer can influence the noise each superconducting circuit , and thus each qubit , is subj ect to . This illustrates why a detailed noise characteri zation needs to take into account the individual qubit and the individual gate .

[0042] In the context of quantum mechanics , in particular in the context of QC model , symmetries are represented by operators that commute with the operator representing the time evolution of the qubits , that is , the quantum circuit . In more detail , i f the time evolution can be represented by an operator U, an operator S is said to be a symmetry of U i f the two operators commute , that is [U,S] = US - SU = 0, which is equivalent to USU^ = S since the time evolution is unitary : U^U = 1. This can be understood in the sense that the order in which the two operators are applied does not influence the outcome and thus means that a measurement of S before and after the evolution U yields the same result . As a consequence , S is said to be a conserved quantity since an eigenstate of this operator remains an eigenstate with the same eigenvalue before and after applying the evolution operator U and the operator S spans one or more subspaces to which the time evolved final state is restricted to .

[0043] In view of the above , it can be understood that i f an error occurs during the quantum circuit, the actual time evolution is not represented by the operator U but by a faulty version thereof . This error may lead to the state of the qubits leaving the subspace since the faulty version of the time evolution may not commute with the symmetry S . As this means that the quantity conserved under the time evolution U is no longer conserved, such an error can be detected by measuring the symmetry S . As a consequence of this , a result obtained by such a faulty execution of the quantum circuit can be discarded, thus improving the accuracy of the overal l computation . This concept is also referred to as "post selection by symmetry veri fication" since based on a veri fication of whether a symmetry is broken by the execution of a quantum circuit ( i . e . , measuring the symmetry operator S) a post selection is carried out ( i . e . , based on the result of the symmetry veri fication incorrect results are discarded) .

[0044] Using the above , the following describes embodiments of the present disclosure in detail .

[0045] Fig . 1 shows a flow chart for a method of quantum error reduction, more in detail a flow chart for a computer- implemented method of quantum error reduction applied to a quantum circuit to be executed on a quantum processing unit , wherein the quantum circuit has one or more symmetries , each represented by a symmetry operator , and the quantum processing unit is provided with a noise characteri zation, each operator element of the noise characteri zation represented by a noise operator, the method comprising the steps of : identi fying at least noise operators that anticommute with at least one symmetry operator as detectable errors ( S 100 ) ; identi fying noise operators not comprised in the detectable errors as non- detectable errors ( S200 ) ; performing a first error reduction process of reducing the non-detectable errors ( S300 ) ; and performing a second error reduction process of reducing the detectable errors ( S400 ) .

[0046] Here , "quantum error reduction" may be considered to encompas s quantum error correction and quantum error mitigation . That is , quantum error reduction may be seen as the umbrella term for any techniques addressing errors in quantum computing .

[0047] A quantum circuit having a symmetry may be considered to mean that executing the quantum circuit on a quantum processing unit conserves a quantity of the quantum processing unit , the quantity being associated with the symmetry .

[0048] That is to say, an error- free execution of the quantum circuit on the quantum processing unit conserves the quantity, while errors induced by noise either break this symmetry, in which case the quantity is no longer conserved but changes or may also not break this symmetry, in which case the quantity remains conserved, but nevertheless lead to an incorrect execution of the quantum circuit . To avoid any ambiguity, there are errors that do not break the symmetry and both types o f errors are addressed in the present disclosure . Indeed, the present invention may be understood to address all errors , but to di f ferentiate between those that break a symmetry and those that do not break any symmetry in the way these errors are reduced, thereby arriving at a more ef ficient quantum error reduction method .

[0049] A noise characteri zation may be understood as comprising of operator elements , for example Kraus operators , and weights , for example weights associated with the Kraus operators . In speci fic examples , these Kraus operators may be obtained from Lindblad operators , for example if a Pauli-Lindblad model is used .

[0050] It is noted that performing the first error reduction process and / or performing the second error reduction process is part of the execution of the quantum circuit . In other words , the first and / or second error reduction process may be understood as elements incorporated into the execution of the version of the quantum circuit not taking into account any errors at all . Herein, " incorporated" is not to be understood that these steps necessarily have to take place during the execution of the quantum circuit but may instead also take place after execution of the quantum circuit .

[0051] The first and second error reduction process may di f fer from each other, in particular they may di f fer from each other in terms the scaling of the number of additional quantum computer runs required to mitigate the error . For example , the scaling with respect to the number of qubits of the quantum processing unit and / or the amount of noise in the quantum processing unit . Speci fically, the second error reduction process of reducing detectable errors may scale better than the first error reduction process of reducing non-detectable errors .

[0052] Herein, performing a process , including the first and the second error reduction process , but not limited thereto , may also include " sending instructions to a quantum processing unit to carry out the necessary steps" . That is , the quantum error reduction method may relate to a method in which the steps are performed on a quantum computer comprising the quantum processing unit , or a method in which the steps are performed partially on a quantum computer comprising the quantum processing and partially on a classical computer, or a method in which all steps are performed on a classical computer, these steps being limited to instructing a quantum computer comprising the quantum processing unit to perform experiments to provide result and evaluating these results to reduce the errors .

[0053] In other words , since the quantum error reduction is part of executing a quantum circuit and since performing a quantum circuit can be seen as a result of data processing measurements results obtained by performing experiments on the quantum processing unit , the present invention does not necessarily involve the quantum processing unit but may also be implemented through interacting with the same by sending instructions / input to the quantum processing unit and receiving results / output of , for example , the first / second process , from the quantum processing unit .

[0054] Fig . 2 shows an illustration of detectable and non-detectable errors , in particular for the speci fic case that any error can be attributed to either of these two categories . Speci fically, Fig . 2 shows an illustration of possible evolutions of an initial quantum state with the full Hilbert space . In more details , in Fig . 2 it is assumed that the ideal evolution U is such that a set of symmetries {S leading to a set of conserved quantities is present and thus a valid subspace can be defined to which the evolution of states starting with this subspace should be confined due to these symmetries . Indeed, as shown by the solid black arrow depicting the ideal evolution U, any initial state within this subspace should lead to a final state within this subspace .

[0055] The presence of errors , however, makes it so that the actual final state may be a state di f ferent from the " final state" depicted in Fig . 2 . Instead, as shown by the dashed arrows , faulty evolution may take various forms and in particular lead to various types of faulty final state .

[0056] The first type of faulty final states is shown in the top right of Fig . 2 . Here , the faulty evolution leads to a faulty final state outside of the valid subspace . This means that the symmetry is broken and the quantities that are conserved by a correct evolution are not conserved . Hence , an ( ancillary) measurement of these symmetries will indicate that the final state is not with the valid subspace and thus an error must have occurred . Hence , an error that modi fies the ideal evolution U such that the final state is outside of the valid subspace is a detectable error .

[0057] As an example , in a system with the parity of the qubits as a conserved quantity, that is , the symmetry is given by the operator 0^ erf an error induced by a Pauli-X operator <5Xacting on a single qubit changes the parity of the qubits and thus such an error would be a detectable error .

[0058] The second type of faulty final states is shown in the bottom center of Fig . 2 . Here , a faulty evolution took place since the indicated final state is di f ferent from the final state of the ideal evolution U, however, the state remains within the valid substate . In other words , an error occurred that did not break the symmetries , i . e . , did not violate the conserved quantities .

[0059] Taking again the above example of the parity of the qubits being a conserved quantity, an error induced by a Pauli- Z operator crzacting on a single qubit does not change the parity of the qubits and thus such an error would be a non-detectable error . In a similar manner, two Pauli-X operators <5Xacting on two neighboring qubits does not change the parity of the qubits either and thus such an error would be a non-detectable error as well . Further, since a quantum circuit typically includes many quantum gates , more than one error might occur during the execution of a quantum circuit . In such a case , it might happen that two errors occur, and both change the value of the conserved quantity, but in a manner that the first error leads the state outside of the valid subspace and the second error leads the state into the valid subspace again . This i s illustrated by the third dashed error in Fig . 2 connected this faulty final state outside the valid subspace with another faulty final state inside the valid subspace .

[0060] In this case , the overall final state is due to a non- detectable error since it is within the valid subspace , however, i f the conserved quantity is check for the intermediate state , an error would be detected .

[0061] The third type of error illustrates that it is relevant at which stage of the quantum circuit the symmetry veri fication is carried out and that the frequency of the symmetry veri fication may impact the quantum error reduction as well , in particular that a more frequent symmetry veri fication may increase the number of errors detected .

[0062] The above discussion underlines that "detectable error" and "non-detectable error" do not refer to the question whether an error is per se detectable , but rather whether it is detectable on the basis of a symmetry veri fication . "Detectable error" may thus mean "error detectable by symmetry veri fication" and "non-detectable error" may mean "error not detectable by symmetry veri fication" .

[0063] While the above examples have been made with reference to a single symmetry and thus a single conserved quantity, it is stressed that the present disclosure is not limited to the same . In fact , the present invention can be applied to any scenario in which one or more symmetries are present . While in the case of "detectable or non-detectable" i s straightforward in the sense that it directly corresponds to "break the symmetry or maintain the symmetry" , the case of two or more symmetries is , as indicated above , a bit di f ferent : I f any of the two or more symmetries is broken, it can be inferred that an error has occurred and hence such an error is a "detectable error" . I f none of the two or more symmetries , it cannot be inferred based on symmetry veri fication that an error has occurred and thus an error that does not break any of the symmetries are "non-detectable errors" . Thus , having more than one symmetry substantial increases the number of errors that can be detected by symmetry veri fication and thus may ampli fy the advantages obtained by the present invention .

[0064] It is noted that the above explanations are based on the assumption that any error that may occur is either a detectable error or a non-detectable error . This assumption is valid for a maj ority of systems , especially for systems of practical relevance , such as systems in which the noise operators and the symmetry operators can be represented using Pauli operators by means of Pauli twirling / Randomi zed Compiling .

[0065] On the other hand, i f this situation is not given (which may be equivalent to the situation that any pair of noise operator and symmetry operator cannot be classi fied into "commuting with each other" or "anticommuting with each other" ) , there is an additional type of error, namely, partially (non- ) detectable errors . These are errors which are neither completely detectable by means of symmetry veri fication nor completely undetectable by means of symmetry veri fication . This can be understood when considering that a possible noise operator may be a linear combination of two noise operators , a first one leading to a detectable error and a second one leading to a non-detectable error . Because of this , a state subj ect to this "combined" noise operator will be a superposition between a state maintaining the symmetry and a state breaking the symmetry and thus a symmetry veri fication, e . g . , a measurement , will only be able to detect the error in some instances .

[0066] Partially (non- ) detectable errors may thus be seen as a special case between detectable errors and non-detectable errors since there relation to the symmetry inherent to the system is not as clear cut as for the other two cases . Nevertheless , these partially (non- ) detectable errors may, since they can be detected, be used within the present disclosure as a "detectable error" within the meaning of methods discussed herein . On the other, since they are not "completely detectable" , they may also be used within the present disclosure as a "non-detectable error" within the meaning of methods discussed herein .

[0067] In view of the above , in a preferred embodiment , the non- detectable errors may comprise noise operators that commute with the one or more symmetries .

[0068] This can also be understood as follows : As discussed above in relation to steps S 100 and S200 , the detectable errors may be characteri zed with reference to those that anticommute with at least one symmetry operator, thus being "always detectable" errors , and the non-detectable errors may be defined as the "negative" , that is , the remaining errors .

[0069] This characteri zation of the non-detectable errors can be further speci fied by characteri zing them not only with reference to the detectable errors , but with respect to their own properties , namely that the non-detectable errors comprise at least those operators that commute with all of the symmetry operators .

[0070] These noise operators may be understood as corresponding to the errors that are "completely" non-detectable since the fact that these noise operators commute with all of the symmetries present means that they will not move any state out of its symmetry subspace .

[0071] Further, in a preferred embodiment , the detectable errors may further comprise at least one noise operator that does not anticommute with any one of the one or more symmetries and does not commute with at least one symmetry .

[0072] This at least one operator may be understood as corresponding to a partially (non- ) detectable error . Speci fically, the first aspect of the above may distinguish these operators from those corresponding to the "always" detectable errors since it excludes that the operators anticommute with any symmetry . The second aspect of the above may distinguish these operators from those corresponding to "never" detectable errors since it excludes that the operator commutes with all symmetries .

[0073] In a preferred embodiment , the non-detectable errors may further comprise at least one noise operator that does not anticommute with any one of the one or more symmetries and does not commute with at least one symmetry .

[0074] It can be understood that these characteri zations correspond to each other and only di f fer in the determination whether the partially (non- ) detectable errors should be treated as detectable errors , and thus subj ect to the second error reduction process , or should be treated as non-detectable errors , and thus subj ect to the first error reduction process .

[0075] It is further noted that it is possible that some of the partially (non- ) detectable errors are treated as detectable errors while others are treated as non-detectable errors . For example , i f such a partially (non- ) detectable error is due to a superposition of noise operators corresponding a detectable and a non-detectable error, the weights associated to these di f ferent types of errors may be indicative whether the error should be treated as detectable or non-detectable in the context of the quantum error reduction method of the present disclosure .

[0076] In a preferred embodiment of any of the above methods , the wherein the first error reduction process may be an error mitigation process , preferably a probabilistic error cancelation ( PEC ) process .

[0077] As explained also elsewhere , an error mitigation process may be understood as a process directed at mitigating the error during the execution of the quantum circuit rather than correcting the error after executing the quantum circuit .

[0078] One established process is PEC in which, based on a noise characteri zation, additional quantum gates are added probabilistically to the quantum circuit to be executed such that these additional gates cancel the error induced into the quantum circuit execution .

[0079] In a preferred embodiment of any of the above methods , the second process may be an error mitigation process , preferably a post selection process and / or a symmetry veri fication process .

[0080] As explained also elsewhere , a symmetry veri fication process may be understood as a process of determining (veri fying) whether executing the quantum circuit on the quantum processing unit maintained all of the symmetries or whether executing the quantum circuit on the quantum processing unit violated at least one of the symmetries .

[0081] Further, post selection may be considered as a process o f selecting the results after executing the quantum circuit on the quantum processing unit . This may be based on some ancillary measurement such as the above discussed symmetry veri fication . This particular example may thus be understood as "post selection based on symmetry veri fication . In a preferred embodiment of any of the above methods , the second process is an error correction process , preferably based on a stabili zer quantum error-correcting code .

[0082] As explained also elsewhere , an error correction process is directed at correcting the error rather than mitigating the error . In this regard, error correction may be seen as similar to post selection in that the process takes place at the end of or after the execution of the quantum circuit . Di f ferent from the above considerations regarding post selection, error correction is oftentimes more elaborate compared to post selection : While post selection mostly analyzes whether it can be inferred that a result is incorrect and discards it in such a case , error correction aims at identi fying the cause of the error such that the error can be corrected such that an incorrect result can be turned into a correct result . From this , it can be understood that error correction may be a more intricate process compared to error mitigation but may yield bigger improvements on the precision of the results .

[0083] It is further noted that an error correction process may be particularly advantageous i f more than one symmetry is present and some form of error correction process may require more than one symmetry to properly identi fy ( and subsequently correct ) the origin of a detected error .

[0084] In a preferred embodiment of any o f the above methods , the one or more symmetries may each be represented by a Pauli string . A Pauli string may refer to a Kronecker product of a plurality of Pauli operators .

[0085] In a preferred embodiment of any of the above methods , the noise characteri zation may be represented by a Markovian noise model .

[0086] In the Markovian noise model , the noise can be written as an operator £[p] = (1 - Si Pi) p + Si PiPipPi and then the inverse of the noise (up to second order in the coefficients p as 8x[p] =

[0087] (1 + SiPi)p -liPiPipPi-

[0088] This formulation of the inverse noise may mean that if operators are the source of non-detectable error, that is, respect the one or symmetries, (probabilistically) cancelling these errors with the same operators also respects the symmetry. This may ensure that in this case the first process does not interact destructively with the second process.

[0089] In a further preferred embodiment, the noise characterization is represented by a Pauli-Lindblad model, preferably by a sparse Pauli-Lindblad model.

[0090] This may be equivalent to say that each operator element of the noise characterization is represented by a Pauli string. The complete noise characterization may be obtained by the operator elements and their associated weights / probabilities . Further, if the operators P, are Pauli operators, this channel can be referred to as a Pauli error channel.

[0091] A noise characterization and the one or more symmetries admitted a representation in terms of Pauli operators may be particularly advantageous since Pauli operators either commute or anticommute with each other, which may be particularly advantageous for symmetry verification.

[0092] In a preferred embodiment of any of the above methods, it may be possible to transpile the quantum circuit into Clifford layers of multi-qubit gates and non-Clifford layers.

[0093] In a preferred embodiment, a variation of the quantum circuit in which gates of the non-Clifford layers are replaced with their complex conjugate may have the same one or more symmetries. That is, the resulting circuit may still preserve the same symmetries even if any subset of the non-Clifford gates within the non-Clifford layers is complexly conjugated. In a preferred embodiment, it may be possible to write each non-Clifford layer as exp(iaP), where a is a real number and P is a Pauli operator. The Pauli operator can be a single-qubit Pauli operator or a multi-qubit Pauli operator. Here, a singlequbit Pauli operator may be understood as a Pauli operator that acts on a single qubit, while a multi-qubit Pauli operator may be understood as a Pauli operator that acts on at least two qubits. The term multi-qubit Pauli operator may commonly be referred to as a "Pauli string".

[0094] In a further preferred embodiment, the non-Clifford layers may consist of single-qubit gates. Indeed, any operator of the form exp(iaP) can be decomposed into a Clifford layer followed by a non-Clifford single-qubit rotation, followed by a further Clifford layer, also explained in arxiv. org / abs / 2303.04498.

[0095] Further, transpiling (also: compiling) a quantum circuit may be considered as a process of converting a high-level quantum algorithm, which may be represented by a quantum circuit, into a specific set of elementary quantum gate operations that can be executed on a physical quantum computer hardware, such as a quantum processor, a quantum processing unit or the like.

[0096] A Clifford layer may mean a layer of Clifford gates and / or mean a layer of gates, wherein the gates of said layer are an element of the Clifford group. Importantly, that mean that not every gate of the Clifford layer needs to be an element of the Clifford group.

[0097] The term "multi-qubit" may mean two-qubit or more and thus multi-qubit gate refers to a gate acting in parallel on two or more qubits.

[0098] Accordingly, a non-Clifford layer may mean a layer of gates, wherein the unitary operator corresponding to the said layer is not an element of the Clifford group. In a preferred embodiment , the non-Cli f ford layers may consist of single-qubit gates .

[0099] In a preferred embodiment of any of the above methods , the method comprises a step of zero noise extrapolation ( ZNE ) .

[0100] As explained also elsewhere , ZNE can be understood as the concept of ampli fying the noise in the quantum circuit executed on a quantum processing unit such that several results with di f ferent noise levels are obtained . Using these results , one can perform extrapolation to obtain the result of zero noise .

[0101] In a preferred embodiment , the step of zero noise extrapolation may include : performing the first error reduction process of reducing the non-detectable errors and the second error reduction process of reducing the detectable errors at di f ferent noise strengths , resulting in a set of noisy results of the quantum circuit , and extrapolating a noiseless result from the set of noisy results .

[0102] In a further preferred embodiment , the step of zero noise extrapolation may include : performing the second error reduction process of reducing the detectable errors at di f ferent noise strengths , resulting in a set of noisy results of the quantum circuit , and extrapolating a noiseless result from the set of noisy results or the step of zero noise extrapolation may include : performing the second error reduction process of reducing the non-detectable errors at di f ferent noise strengths , resulting in a set of noisy results of the quantum circuit , and extrapolating a noiseless result from the set of noisy results . In other words , the zero noise extrapolation can be applied to both, the detectable errors as well as the non-detectable errors .

[0103] In a preferred embodiment of any of the above methods , the second process may be performed only at the end of the quantum circuit , may be performed after each layer of the quantum circuit , or may be performed after at least one layer of gates of the quantum circuit that preserves at least one of the one or more symmetries .

[0104] It can be understood that performing the second process only at the end of the quantum circuit may involve the least number of additional steps .

[0105] It can further be understood that while performing the second process more often may provide more improvement and require additional steps to be performed .

[0106] In this context , it may be advantageous to take into consideration that any quantum operation, including those constituting the quantum error reduction, can be performed only with finite accuracy . I f the second process could be carried out perfectly, then performing this as often as possible would likely be a very advantageous approach . In any practical setting, the second process however involves measurements and extra quantum gates , both of which may introduce additional errors since any quantum gate / quantum operations may be faulty . Thus , there might be the scenario that the probability that an additional step of quantum error reduction introduces an error is larger than the probability that an actual error occurs in the quantum circuit . In this case , it may be advantageous to reduce the frequency of the second process .

[0107] Fig . 3 shows a comparison of the relative cost ( left ) and the error ( right ) of methods according to the present invention with conventional approaches of quantum error reduction . The underlying model of these results is the one-dimensional t-J Hamiltonian representing a one-dimensional chain of fermions with quadratic nearest neighbor interaction . In this Hamiltonian, the number of fermions is conserved . Further, by means of a Jordan-Wigner-trans formation this Hamiltonian can be mapped into a spin- 1 / 2 Hamiltonian, i . e . , a qubit Hamiltonian . The conserved quantity is then mapped onto the above-mentioned parity represented by the operator 0, 0"? . This Hamiltonian is simulated for three qubits using a Trotteri zation with a Pauli noise applied for each Trotter step with noise strength y = 10-2, t = 1.00, J = 2.00 and a Trotter step si ze of e = 0.750.

[0108] The left-hand side shows the cost C as a function of the number of Trotter steps while the right-hand side shows the error l<oest.> - <0>l for 0 = (5Z. The cost is defined as and therefore determines how many more runs of the quantum computer are needed in order to determine the mitigated observable 0estwith the same statistical accuracy as in the unmitigated case .

[0109] From the left plot it can be seen that the cost of methods according to the present invention lie between the costs of quantum error reduction using only post selection based on symmetry veri fication, which has - as expected - a lower cost , and the costs of quantum error reduction using only probabilistic error cancelation, which has - also as expected - a higher cost .

[0110] From the right plot it can be seen that methods according to the present invention also strike a good balance in terms of the remaining error after the quantum error reduction is performed . As expected, the far more costly PEC achieves a smaller error than the methods according to the present invention while post selection based on symmetry veri fication is able to reduce far less errors compared to the methods according to the present invention . Fig. 4 shows comparisons of the mean-square-error of methods according to the present invention with conventional approaches of quantum error reduction. Here, the mean-square- error is the sum of the squared bias and the squared variance, thus, very closely related to Fig. 4.

[0111] The left side of Fig. 4 shows results in which the second error reduction process, e.g., post selection, is performed after each Trotter step, while the right side of Fig. 4 shows results in which the second error reduction process is performed only at the end of the circuit. The rows of the left and right side of Fig. 4 correspond to each other and correspond to different average numbers of errors as indicated. As indicated in the legend, the black dot-dashed line represents post selection, the grey dashed line represents PEC and the dark grey dotted line represents methods according to the present invention. Further, the grey column present in each plot indicates the range of number of shots in which the methods according to the present disclosure shown an improvement compared to the conventional approaches, i.e., show the lowest mean-square- error of all three shown methods.

[0112] The shown behavior is in line with the discussed expectations that PEC tends to be able to reduce the errors to the largest extent if many resources are available, that PS is able to reduce errors most efficiently if only few resources are available and that the present invention is able to find a balance between the two methods that shows an improvement over both methods for a large range of practically relevant number of shots.

[0113] Further, when comparing the left side with the right side of Fig. 4, it becomes clear that performing the second process, i.e., PS more frequently improves the advantage provided by the present invention over conventional methods. In a further preferred embodiment of the presented invention, there is provided a quantum computer having a quantum processing unit , the quantum computer configured to implement a method according to an embodiment of the present invention .

[0114] This quantum computer may, for example , be able to receive , as an input , a quantum circuit to be executed by the quantum computer and then perform, based on appropriately defined routines in line with the above , a quantum error reduction method . In other words , the quantum computer can perform, given the quantum circuit , the quantum error reduction method, that is , one may say that the quantum computer performs the quantum error reduction method autonomously .

[0115] There are no particular limitations regarding the possible hardware platforms on which the present invention may be implemented . Quantum processing unit based on superconducting circuits / superconducting qubits are a clear use case , in particular as the above cited papers have demonstrated that conventional cycle benchmarking can be reali zed on this platform . Nevertheless , an adaption to any platform such spin qubits , ultra-cold atoms and the like is possible as the above discussion is platform independent .

[0116] In a further embodiment according to the present invention, there is provided a classical computer having a classical processing unit , the classical computer configured to implement a quantum error reduction method according to an embodiment of the present invention .

[0117] This corresponds to the above discussion that the present invention may be implemented by a classical computer controlling a quantum computer, providing input to the quantum computer and receiving an output from it . In this scenario , the quantum computer could be seen as an external ( technical ) entity controlled by a classical computer . In a still further embodiment according to the present invention, there is provided a system of a quantum computer according to an embodiment of the present invention and a classical computer according to an embodiment of the present invention . In other words , this system comprises the quantum computer as well as the classical computer . In such a system, the various steps involved in performing the quantum error reduction method may be shared and / or distributed among the classical computer and the quantum computer . They may in particular be distributed in a manner that allows ef ficient performance of said steps .

[0118] In summary, the present invention provides quantum error reduction methods exploiting the presence of one or more symmetries in the underlying quantum operations such that the quantum circuit admits conserved quantities in a manner that allows for a more ef ficient quantum error reduction by reducing those errors detectable in view of the symmetries and those not detectable in view of the symmetries in a di f ferent manner .

[0119] Advantageously, this allows to reduce errors more ef ficiently without losing the possibility to correct all errors since errors detectable by reference to symmetries can be reduced with less resources compared to the more general quantum error reduction methods .

[0120] Consequently, the present invention can be understood as providing a quantum error reduction method that allows to address those errors cost-ef f iciently that can be addressed in such a cost-ef ficient manner and addresses the remaining errors using more general methods .

[0121] Further, the present invention is not particular limited to a quantum processing unit topology, a quantum processing unit plat form / technology or the like , and thus can find application across the whole spectrum of quantum computing technologies , in particular can implemented whenever conventional quantum error reduction methods can be implemented and the quantum circuit to be executed admits conserved quantities .

Claims

Claims1 . A method of quantum error reduction applied to a quantum circuit to be executed on a quantum processing unit , wherein the quantum circuit has one or more symmetries , each represented by a symmetry operator, and the quantum processing unit is provided with a noise characteri zation, each operator element of the noise characteri zation represented by a noise operator, the method comprising the steps of : identi fying at least noise operators that anticommute with at least one symmetry operator as detectable errors ; identi fying noise operators not comprised in the detectable errors as non-detectable errors ;- performing a first error reduction process of reducing the non-detectable errors ; and- performing a second error reduction process of reducing the detectable errors .2 . The method according to claim 1 , wherein the non- detectable errors comprise noise operators that commute with the one or more symmetries .3 . The method according to claim 1 or 2 , wherein the detectable errors further comprise at least one noise operator that does not anticommute with any one of the one or more symmetries and does not commute with at least one symmetry .4 . The method according to any one of claim 1 to 3 , wherein the non-detectable errors further comprise at least one noise operator that does not anticommute with any one ofthe one or more symmetries and does not commute with at least one symmetry.

5. The method according to any one of claims 1 to 4, wherein the first error reduction process is an error mitigation process, preferably a probabilistic error cancelation process; and / or wherein the second error reduction process is an error mitigation process, preferably a post selection process and / or a symmetry verification process; and / or wherein the second process is an error correction process, preferably based on a stabilizer quantum errorcorrecting code.

6. The method according to any one of claims 1 to 5, wherein the one or more symmetries are each represented by a Pauli string.

7. The method according to any one of claims 1 to 6, wherein the noise characterization is represented by a Markovian noise model; wherein preferably the noise characterization is represented by a Pauli-Lindblad model, preferably by a sparse Pauli-Lindblad model.

8. The method according to any one of claims 1 to 7, wherein the quantum circuit can be transpiled into Clifford layers of multi-qubit gates and non-Clifford layers ;wherein preferably a variation of the quantum circuit in which gates of the non-Clifford layers are replaced with their complex conjugate has the same one or more s ymme tries.

9. The method according to claim 8, wherein each nonClifford layer can be written as exp(iaP), where a is a real number and P is a Pauli operator, in particular a single-qubit Pauli operator or a multi-qubit Pauli operator; wherein in particular the non-Clifford layers consist of single-qubit gates.

10. The method according to any one of claims 1 to 9, wherein the method comprises a step of zero noise extrapolation .

11. The method according to claim 10, wherein the step of zero noise extrapolation includes- performing the first error reduction process of reducing the non-detectable errors and the second error reduction process of reducing the detectable errors at different noise strengths, resulting in a set of noisy results of the quantum circuit, and- extrapolating a noiseless result from the set of noisy results; or wherein the step of zero noise extrapolation includes- performing the second error reduction process of reducing the detectable errors at different noisestrengths , resulting in a set of noisy results of the quantum circuit , and- extrapolating a noiseless result from the set of noisy results . or wherein the step of zero noise extrapolation includes- performing the second error reduction process of reducing the non-detectable errors at di f ferent noise strengths , resulting in a set of noisy results of the quantum circuit , and- extrapolating a noiseless result from the set of noisy results .12 . The method according to any one of claims 1 to 11 , wherein the second process is performed only at the end of the quantum circuit , is performed after each layer of the quantum circuit , or is performed after at least one layer of gates of the quantum circuit that preserves at least one of the one or more symmetries .13 . A quantum computer having a quantum processing unit , the quantum computer configured to implement the method according to any one of claims 1 to 12 .14 . A classical computer having a classical processing unit , the classical computer configured to implement the method according to any one of claims 1 to 12 .15 . A system comprising a quantum computer according to claim 13 and a classical computer according to claim 14 .

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