Efficient distillation of quantum states of light

Abstract

Methods and system of reducing the partial distinguishability error of quantum state of light are described, wherein the methods include providing partially indistinguishable quantum states of light with an initial error to the input of at least one first N-mode linear interferometer, which is configured to transform quantum state of light into an entangled multimode state, the output modes of the first linear interferometer including ancillary modes and a herald output mode; determining measurement values, the determining including measuring the ancillary outputs of the linear interferometer; determining if the measurements values match one of a plurality of predetermined first herald patterns associated with the configured first linear interferometer, each of the first herald patterns being indicative of a state of the first linear interferometer in which, depending on the initial error, the first linear interferometer can generate a quantum state of light with a reduced partial distinguishability error at the herald output; and, generating a herald signal indicative of at least one quantum state of light with a reduced partial distinguishability error at the herald output of the first linear interferometer if the measurements values match one of the plurality of first herald patterns and if the initial error is below a predetermined threshold error associated with the matched first herald pattern.

Description

[0001] NL38224 -Vi / tdEfficient distillation of quantum states of lightTechnical field The disclosure generally relates to the reduction of partial distinguishabilityin quantum states of light, e.g. photons , and in particular, though not exclusively, tomethods and systems for efficiently reducing partial distinguishability errors of quantumstates of light using an optical interferometer and computer program product using suchmethods. Background Generating highly indistinguishable photons is one of the major challengesin realizing optical quantum computers. Photon distillation is a method to reduce errors ofpartial distinguishable photons as described in the article by Marshall Distillation ofindistinguishable photons, Phys. Rev. Lett.129, 213601 (2022). It is a nondeterministicmethod for heralding indistinguishable photons wherein noisy quantum states of light,typically single photons produced by imperfect sources, are probabilistically exchangedfor one quantum state of light with reduced partial distinguishability error. The quality of adistillation protocol is characterized by the error reduction and herald probability that thescheme provides. Here, the herald probability refers to the successful execution of onedistillation run yielding one or more distilled photons with a reduced partialdistinguishability error, in the absence of losses, and the error reduction refers to thedegree in which the error of the resulting quantum state of light is reduced in the event of successful distillation. The distillation step is mediated by multiphoton interference, which can be repeated indefinitely to yield near-perfect indistinguishable photons. However, each yielded near-perfect photon comes at the expense of some partially distinguishable photons. These two figures of merit can be combined into one: the number of photons which is required to achieve a constant reduction of error, or equivalently, the error reduction available with a fixed number of photons. The photon distillation protocol proposed by Marshall uses a combination ofstandard linear optical networks, optical switches and feedforward electronics to increasephoton indistinguishability to arbitrary accuracy by sacrificing ancillary photons. Ideally, the number of photons that need to be sacrificed to achieve a certain error reduction is kept at a minimum. Via an extensive numerical search, the optimal networks for three- mode and four-mode networks were found. Using these networks in the distillation protocol reduces the error by a factor 1 / 3, with a success rate of 1 / 3. The success rate is determined by the probability of measuring the heralding pattern that reduces the error and can be directly related to the resources (the number of photons) required to achievesuccessful distillation. The 4-mode interferometer protocol reduces the error even furtherby a factor of 1 / 4. This would be very useful if it were not for the fact that the success rate decreases to 1 / 4. The average number of photons required to successfully distil the photon would be 3*3 for the 3-modeand 4*4 for the 4-mode interferometer. This suggeststhat an adequate n-mode system that could reduce the error by a factor of 1 / ^ wouldhave a success probability of 1 / ^ and would therefore need ^^ photons to performsuccessful distillation. The fact that the amount of resources required to realise n-mode photon distillation scales with ^^suggests that it is not useful to employ interferometerswith a greater number of modes for photon distillation.Hence, from the above it follows that there is a need in the art for efficientphoton distillation schemes for generating photons having a reduced partialdistinguishability error. Summary As will be appreciated by one skilled in the art, aspects of the present invention may be embodied as a system, method or computer program product. Accordingly, aspects of the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a "circuit," "module" or "system." Functions described in this disclosure may be implemented as an algorithm executed by a microprocessor of a computer. Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied, e.g., stored, thereon. Aspects of the present invention are described below with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor, in particular a microprocessor or central processing unit (CPU), of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer, other programmable data processing apparatus, or other devices create means for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks. These computer program instructions may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium produce an article of manufacture including instructions which implement the function / act specified in the flowchart and / or block diagram block or blocks. The computer program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions / acts specified in the flowchart and / or block diagram block or blocks. Additionally, the Instructions may be executed by any type of processors, including but not limited to one or more digital signal processors (DSPs), general purpose microprocessors, application specific integrated circuits (ASICs), field programmable logicarrays (FP- GAs), or other equivalent integrated or discrete logic circuitry.The flowchart and block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present invention. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and / or flowchart illustrations, and combinations of blocks in the block diagrams and / or flowchart illustrations, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions. In an aspect, the embodiment in this disclosure relate to a method ofreducing a partial distinguishability error of a quantum state of light comprising: providingpartially indistinguishable quantum states of light with an initial error to the input of at leastone first N-mode linear interferometer, which is configured to transform the quantumstates of light into an entangled multimode state, the N outputs of the first linearinterferometer including N-1 ancillary outputs and a herald output; determining photonnumber measurement , the determining including measuring quantum states of light at theancillary outputs of the linear interferometer; determining if the measurement valuesmatch one of a plurality of predetermined first herald patterns associated with theconfigured first linear interferometer, each of the first herald patterns being indicative of astate of the first linear interferometer in which, depending on the initial error, the first linear interferometer can generate a quantum state of light with a reduced partialdistinguishability error at the herald output.Hence, the inventors found that for certain interferometer implementationsmultiple herald patterns exist, wherein each herald pattern is associated with a certainerror reduction of the initial error of partially indistinguishable quantum states of light thatare provided to the input of the interferometer. The consequence of the existence ofmultiple valid heralding patterns is that the number of quantum states of light required toachieve a constant error reduction is substantially reduced compared to the state of the art. Whereas previously, the quadratic scaling was required to achieve a constant ^ fractional improvement in indistinguishability, i.e. ^^∝ ^^, where ^ is the number ofphotons, now linear scaling can be achieved in the limit of asymptotically many photons, ^ i.e. ^^∝ ^.In an embodiment, the method may further comprise generating a heraldsignal indicative of at least one quantum state of light with a reduced partialdistinguishability error at the herald output of the first linear interferometer if themeasurements values match one of the plurality of first herald patterns and if the initialerror is below a predetermined threshold error associated with the matched first herald pattern. In an embodiment, the method further comprises: controlling an opticalswitch at the herald output of the first linear interferometer based on the herald signal.In an embodiment, the method may further comprise: providing the at leastone quantum state of light with the reduced partial distinguishability error to a secondlinear interferometer. In an embodiment, the at least one quantum state of light with the reducedpartial distinguishability error may be provided to the second linear interferometer if thereduced partial distinguishability error of the quantum state of light at the herald output ofthe first linear interferometer is smaller than a target error. In an embodiment, the second linear interferometer is configured totransform quantum states of light into an entangled multimode state, which has ancillaryoutputs and a herald output, and which is associated with one or more predetermined second herald patterns, each of the second herald patterns being indicative of a state of the second linear interferometer in which, depending on the reduced partialdistinguishability error of the quantum state of light, the second linear interferometer cangenerate a quantum state of light with a further reduced partial distinguishability error atthe herald output of the second linear interferometer. In an embodiment, the number of modes N of the first linear interferometer may be larger than 4. In an embodiment, the photon states may be single photon states, multi- photon states, squeezed states or optical cat states. In an embodiment, the first linear interferometer may be configured basedon a transformation matrix for performing an optical transformation comprising one or more forbidden outcomes. In an embodiment, the first linear interferometer may be configured basedon a transformation matrix for performing an optical transformation wherein the transformation matrix exhibits symmetry in at least part of the rows and / or columns of the transformation matrix. In an embodiment, the first linear interferometer may be configured toperform a Fourier transform or a Hadamard transform. In an embodiment, the first linear interferometer may be implemented asprogrammable universal multiport interferometer.In an embodiment, the configuring of the first linear interferometer may be based on a transformation matrix. In an embodiment, the transformation matrix may be based on a product of^^,^ matrices, a ^^,^ matrix defining a lossless beam splitter between channel ^ and ^with reflectively ^ and phase shift ^.In an embodiment, the reflectivity and the phase shift parameters of the^^,^ matrices may be used to configure the first linear interferometer.In a further aspect, the embodiments may relate to a system for of reducinga partial distinguishability error of a quantum state of light comprising: at least a first N-mode linear interferometer, which is configured to transform quantum states of light intoan entangled multimode state, preferably the first linear interferometer being configured toperform a Fourier transform or a Hadamard transform, the N outputs of the first linearinterferometer including N-1 ancillary outputs and a herald output; an optical sourcemodule configured to generate partially indistinguishable quantum states of light with aninitial error and providing the partially indistinguishable quantum states of light to the inputof the first linear interferometer; a detector module for determining photon numbermeasurement values , the detector module being configured to measure the ancillaryoutputs of the first linear interferometerIn an embodiment, the system may comprise a processor configured to:determine if the measurement values match one of a plurality of predetermined first heraldpatterns associated with the configured first linear interferometer, each of the first heraldpatterns being indicative of a state of the linear interferometer in which, depending on the initial error, the first linear interferometer can generate a quantum state of light with areduced partial distinguishability error at the herald output.In a further embodiment, the processor may be configured to generate aherald signal indicative of at least one quantum state of light with a reduced partialdistinguishability error at the herald output of the first linear interferometer, if themeasurements values match one of the plurality of the first herald patterns and if the initialerror is below a predetermined threshold error associated with the matched first herald pattern. In an embodiment, the determination of whether quantum state of light withreduced partial distinguishability is generated at the herald output may be performedbased on a pre-computed table, a decision tree or other decision-making logic whichtakes into account the reduction in partial distinguishability In an embodiment, the determination of whether a quantum state of light with reduced partial distinguishability is generated at the herald output may be based on a computation made after the herald pattern is observed. In an embodiment, the system may further comprise an optical switch atthe herald output of the first linear interferometer.In an embodiment, the processor may be further configured to control the optical switch based on the herald signal. In a further embodiment, the system may further comprise: a second linearinterferometer, which is configured to transform quantum states of light into an entangledmultimode state, which has ancillary outputs and a herald output, and which is associated with one or more predetermined second herald patterns, each of the second herald patterns being indicative of a state of the second linear interferometer in which, dependingon the reduced partial distinguishability error of the quantum state of light, the secondlinear interferometer can generate a quantum state of light with a further reduced partialdistinguishability error at the photon output mode of the second linear interferometer.In an embodiment, the optical configuration of the second interferometermay be determined by the heralding pattern observed in the first interferometer. In an embodiment, the optical switch at the herald output of the first interferometer may route the quantum state of light either to the output of the system or to a second interferometer, or it may discard it, based on the reduced partial distinguishability error. In an embodiment, the processor may be further configured to provide theat least one photon state with the reduced partial distinguishability error to the input of the second linear interferometer, preferably if the reduced partial distinguishability error of the photon state at the photon output mode of the first linear interferometer is smaller than a target error. The embodiments may also relate to a computer program or suite ofcomputer programs comprising at least one software code portion the software code portion, when run on a computer, being configured for executing the method steps according any of claims. Brief of the Fig. 1 depicts a hybrid computer system including an optical processor anda classical computer; Fig. 2A-2C schematically depict an example of configurable linearinterferometer; FIG.3 depicts a schematic of a photon distillation system according to anembodiment; Fig. 4 depicts a parametric plot of the fraction as a function of ^imperfect single photon states involved of partial distinguishability error ^;Fig. 5A and 5B depict the error reduction and herald success probability fora photon distillation scheme according to an embodiment. Fig. 6 depicts a method for photon distillation according to an embodiment;FIG.7 depicts the error reduction photon-cost scaling for photon distillationschemes as described with reference to the embodiments in this disclosure. Fig. 8 depicts a photon distillation system according to anotherembodiment; Fig. 9 depicts a photon distillation system according to anotherembodiment. of the embodiments Fig. 1 depicts a photonic system 100 including a photonic integrated circuit104 and a processor 106, e.g. a classical computer. The photonic integrated circuit mayinclude a programmable interferometer 108 in which a function can be encoded in termsof optical operations. The programmable interferometer may be associated with acontroller 110 for programming the interferometer and for controlling the programmedinterferometer. In particular, the controller may include a laser and detectors to control thequantum interference process. Further, the classical computer may include softwareand / or hardware modules which are configured to execute software code. This way, thephotonic system can be configured for specific applications as described with reference tothe embodiments in this application. For example, a classical computer may include an encoder module 112configured to encode certain functions, such as Fourier transforms or other transformationfunctions, into the interferometer so that the system can be configured as a photondistillation system for generating heralded photons that have a reduced partialdistinguishability error. A transformation function may be represented by a transformationmatrix for transforming quantum states of light, such as partially distinguishable photons, into an entangled multiphoton state. Furthermore, the classical computer may include aphoton distillation module 114 configured to control the interferometer, optical circuitsassociated with the interferometers and photon sources and photon detectors to executephoton distillation schemes as described with reference to the embodiments in this application. Partially distinguishable photons may be modelled using an orthogonalbad-bit model (OBBM) which models single photon states produced by imperfect sources.Each single photon state ^ may have a pure internal state |^^^ = ^|^^^ + √1 − ^ |^^^,wherein |^^^ represents a shared overlapping mode and |^^^ is a unique error mode suchthat ^^^|^^^ = ^^^ . The Hong-Ou-Mandel (HOM) visibility may be defined as = = ^^ , which provides a measure for a uniform partial distinguishability 0 ≤^ ≤ 1. The assumption of identical mutual partial distinguishability is demonstrated to beappropriate in multiphoton experiments with imperfect sources of sufficient quality. The internal error modes do not contribute to multiphoton interference because of theirorthogonality, thus allowing expansion of the imperfect single photon states ^(^) ∶=|^^^^^^| into equivalent sums of fully indistinguishable and fully distinguishable photons: where ^: : = ^^^|^(^)|^^^ = 1 − ^ is the definition used in this work for the partialdistinguishability error. The embodiments in this disclosure aim to generate states ^(^′)for which ^′ < ^.Considering partially distinguishable photons equivalently as mixtures of indistinguishable and distinguishable photons, some facts about properties of Fouriertransforms in linear quantum optics may be considered. In linear quantum optics, theFourier transform is implemented as a linear transformation acting on the bosonic modal annihilation operators (^^^, ... , wherein the matrix elements of the modal Fourier transform are given by: In an embodiment, this modal transformation may be implemented on amesh of programmable interferometers. An example of a universal multiportinterferometer which may be used in the embodiments described in the application isdepicted in Fig.2A-2C. In particular, Fig.2A depicts a reconfigurable universal multiportinterferometer comprising N input ports and M output ports. The interferometer mayinclude ^ partially distinguishable photons optical sources 204 wherein each opticalsource 206 may be connected to an input port of the interferometer 202 and configured togenerate partially distinguishable single photons. Alternatively, in another embodiment,the optical sources may procedure partially distinguishable squeezed states or optical catstates. The quantum hardware system may further include ^ single photon detectors 208for detecting photons at ^ of the output ports of the interferometer. It is noted that Fig.2Aonly illustrates a non-limiting example of a multiport interferometer that can be used withthe embodiments described in this application. For example, instead of single photon detectors, detection techniques like photon number resolving detection, homodynedetection or heterodyne detection may be used, and various quantum states of light, e.g.single photon, squeezed states, cat states, etc., may be distilled using such a circuit.Fig. 2B illustrates a reconfigurable universal multiport interferometer whichis characterized by matrix ^ which may describe the internal optical structure of theinterferometer that includes beam splitters and phase shifters in terms of matrix elements.As shown in this figure, different optical arrangements are possible including a design byReck at et. Experimental realization of any discrete unitary operator,” Physical ReviewLetters, vol. 73, no. 1, p. 58, 1994 and the design by Clements et al, An optimal design foruniversal multiport interferometers, (https: / / doi.org / 10.1364 / OPTICA.3.001460). Anotheroptical platform for a quantum photonic processor that can be used for implementing theembodiments in this disclosure is described in the articles by Taballione et al, 20-ModeUniversal Quantum Photonic Processor, arXiv:2203.01801v5 and Taballione et al., Auniversal fully reconfigurable 12-mode quantum photonic processor, Mater. QuantumTechnol. 1 (2021) 035002. These documents are hereby incorporated by reference intothis application. Matrix ^ may be decomposed into a product of ^^,^ matrices: ^ = ^herein a ^^,^ matrix may define a lossless beam splitter betweenchannel ^ and ^ with reflectively ^ and phase shift ^ as illustrated in Fig.2C. Theinterference between photons is the backbone of quantum information processing on a linear optical network as described above with reference to Fig.2A-2C. The interference may be described using the Fock basis. This process may be explained based on a simple two-particle interference setup that has two input and two output modes. With the help of the creation operator ^^^, which simply adds a photon to the ^^^ith mode, a general input state can be described (equation 1): here the input modes are numbered 1 and 2 and the creation operators act on thevacuum state |0^. With the help of a unitary matrix (equation 2): depicting the beamsplitter, creation operators for the output modes, ^^^, can be determined (equation 3): where the output state is expressed with operators ^^^and ^^^ acting on the vacuum state(equation 4): which can be rewritten to (equation 5): creating four final terms, each representing a travel option for the photon. For fully indistinguishable photons, photons with identical degrees of freedom, such aspolarization, frequency or arrival time, the creation operators commute ^^ ^ ^^, ^^^ = 0, anddrop from the expression, resulting in the final output state (equation 6): showing the photons must leave the beam splitter together. Note that the creation operator includes the normalization factor that depends on the number of photons in the mode. The latter two terms will not destructively interfere when the photons are not indistinguishable, due to their no longer commuting operators, ^^^^, ^ ^^^ ≠ 0. The probabilityof detecting the |1,1^ output state is therefore dependent on the partial distinguishability ofthe photons. The two-particle interference setup can be expanded to a more general ^ particle description using a linear optical network containing a series of beam splittersand phase shifters. The input state of the system contains ^ photons in ^ modes(equation 7): where the normalization constant, used before in equation 4, is added to correct for the multiplicity of output states. Following the steps as before, the output creation operatorsare a function of the input creation operators and the ^ · ^ size unitary matrix ^ (equation8): The general expression for the output state can be described by combining bothexpressions (equation 9): where ^ indicated the rows and ^ the columns of the unitary matrix. An alternative way toexpress equation 9 is to use the permanent function of the unitary matrix (equation 10): The permanent of a matrix is analogous to the determinant except for the fact that thesign of the product of elements remain positive instead of alternating signs. Thesummation is altered to run over every combination, ^, in which ^ photons can bedistributed over ^ modes, which is a function of the number of photons, ^, and modes, ^, As an example, there are six combinations to distribute twophotons in three modes, i.e. |2,0,0^, |0, 2, 0^, |0, 0, 2^, |1, 1, 0^, |1, 0, 1^, |0, 1, 1^. In that case,the output state can be based on the permanent using the following expression 10: wherein ^^ is a ^ · ^ sub-matrix of ^ with rows corresponding to the input configurationand columns to the output configurations ^. FIG.3 depicts a schematic of a photon state distillation system according toan embodiment. As shown in the figure, the system may include a photonic integratedcircuit, in particular a configurable interferometer 302 with optical inputs 304 and outputs306. The system may be implemented in a photonic system as described with referenceto Fig.1, which comprises classical processor and control electronics, including opticalsources and detectors, to control the photonic integrated circuit. The photonic integratedcircuit may be implemented as an universal multiport interferometer as described withreference to Fig.2A-2C.The interferometer may be configured to implement a transformation matrix^ 3081,2 describing the properties of the photonic integrated circuit. The transformationmatrix may exhibit symmetry in at least part of the rows and / or columns of thetransformation matrix. These symmetries may give rise to a zero-transmission law, i.e. thefact that some outcomes are fully suppressed for fully indistinguishable photons while for partially distinguishable photons the outcome is not fully suppressed. This is described inmore detail in the article by Tichy, et al. "Zero-transmission law for multiport beamsplitters." Physical review letters 104.22 (2010): 220405 and the article by Dittel et al."Totally destructive many-particle interference." Physical Review Letters 120.24 (2018):240404 both documents are hereby incorporated by reference into this application. Theoutcomes which are suppressed according to the zero-transmission law are referred to asforbidden outcomes¸ while their complementary set is referred to as allowed outcomes.In an embodiment, the transformation matrix may be configured to perform a N-mode Fourier transformation FN 3081. In another embodiment, the transformationmatrix may be configured to perform a N-mode Hadamard transformation HN. In furtherembodiments, the transformation matrix may be configured to perform an optical transformation exhibiting one or more forbidden outcomes. The system may include imperfect sources 314 for generating partlyindistinguishable quantum states of light, for example single or multi photon states,squeezed vacuum states such as single mode squeezed vacuum states (SMSVs) oroptical cat states. In an embodiment, the optical sources may be configured to generate aproduct state ^ partial distinguishable quantum states of light, associated witha certain error ^ which are provided to the input of the photonic integrated circuit. Thephotonic integrated circuit may be configured to transform the optical states by a N-modetransformation matrix, for example an N-mode Fourier transform matrix, into an entangledmultiphoton state. At the output 306 of the photonic integrated circuit, ^ − 1 outputs (referredto as ancillary outputs) may be measured by a detection module 310 resulting in N-1photon number measurements values , where the indices refer to themode numbers. As will be explained below in greater detail, for particular measurementsets and for certain initial errors of the input quantum states of light, at least one quantumstate of light with a reduced error ^′ < ^ will be output N (which may be referred as theherald output). If the measurement values match a so-called heralding pattern, then themeasurement values herald a quantum state of light with a reduced error ^′ < ^ at outputN of the photonic integrated circuit. A N-mode transformation matrix may be associatedwith a predetermined set of heralding patterns, wherein for each heralding pattern thephotonic integrated circuit can produce – depending on the initial error ^ - a quantum stateof light with a reduced indistinguishability error ^′ < ^. These heralding patterns andassociated ranges of initial errors ^ for which the N-mode transformation matrix canproduce a quantum state of light with a reduced indistinguishability error (a distilledquantum state of light) can be computed in advance and stored on a storage medium 315, e.g. a database or a table. Aprocessor 318 may be configured to receive measurement values fromthe detection module and use the stored heralding patterns and associated ranges ofinitial errors ^ for which the N-mode transformation matrix can produce a distilled quantumstate of light to control a further device 312, e.g. a photonic switch, which can block(discard) a quantum state of light or a quantum state of light 316 at the herald output.Hence, the detection module may signal the processor that one of the herald patterns has been measured and the processor may decide based on the storedherald patterns and associated ranges of initial errors ^ for which the N-modetransformation matrix can produce a distilled quantum state of light to control the switch topass a distilled quantum state of light with a certain reduced error. The scheme in Fig.3may be used for distillation of different quantum states of light including as squeezedstates or cat states. As will be described below in more detail, the distillation system may bepart of a larger optical system wherein distilled optical states with a reduced error may beinput to a further optical system, e.g. a further distillation system or a photonic quantumcomputer. The photon distillation system reduces physical noise levels to produce(almost) indistinguishable photons, which are indispensable for numerous applications,including but limited to the generation of entangled multi-photon states which are primitives in photonic quantum computing, Quantum Key Distribution protocols that require entangled photon pairs (such as Bell states) and quantum metrology, whichrequire so-called ^00^ states that allow for high-resolution and highly sensitivemeasurements techniques. Below, embodiments of the distillation process, the determination of theheralding patterns and the associated errors are described in more detail. Although theseembodiments are described based on a Fourier transform and photon modes, it issubmitted that the embodiments are not limited thereto and that different quantum statesof light and other transformations which exhibit symmetry in the rows and / or columns ofthe transformation matrix, such as the Hadamard transformation, can also be used for thedistillation schemes described in this application. In state space representation, the imperfect multiphoton state transforms to For given partial measurements in photon number basis, a single photon state may bedistilled. wherein it is assumed that partial distinguishability errors of photons in modes 2, ... , ^cannot be resolved and ∑^ ^^^ ^^ = ^ − 1. For sufficiently low partial distinguishability error^, the input state may be approximated as where the second term in which the distinguishable photon in mode ^ is placed, isjustified due to the symmetry of the Fourier transform. It may be determined in which error range this first-order approximation isvalid: the approximation is valid if ^^(^) ≪ ^, where ^ ^ ^^^^(^): = 1– (1 − ^) – ^^(1 − ^) isthe probability of higher-order errors. The probability of measuring a product state where^ out of ^ photons are indistinguishable is given by where 1 − ^ is the success probability as defined in Eq.1. The zero-error probability isgiven by ^(^, ^) = (1 − ^)^, the one-error probability may be given by ^(^ − 1, ^) =^(1 − ^)^^ and the probability of more-than-one error may be defined as Since the total probability is conserved, the more-than-one error may be defined as To study the implications of photon distillation based on Eq.5, anapproximation may be used which is valid if the probability of having more than one errorin a product state of ^ imperfect photons is (much) smaller than the partialdistinguishability error in one imperfect photon, i.e. ^^(^) ≪ ^ or ≪ 1. Fig. 4 depictsa parametric plot of the fraction as a function of ^ imperfect single photon statesinvolved of partial distinguishability error ^. The figure shows that the validity range of ^shrinks with increasing ^. Within the validity range of the input error, a one-step distillationprotocol may be used to reduce the output error to meet set target requirements. Outsidethe validity range, one has to check first whether the initial error is below threshold forherald measurements to decide which herald measurements actually reduce the error. Insome embodiment, a concatenation (a sequence) of multiple distillation steps may berequired to achieve a set target error.Fig. 5A depicts the error reduction for a photon distillation schemeaccording to an embodiment. In particular, the figure depicts the error reduction for aphoton distillation scheme that is based on a ^ = 5 mode Fourier transform The figureshows that for certain measurements of the ancillary modes, a purified photon state ^(^′)is heralded that has a reduced partial distinguishability error ^′ < ^. The figure shows thatone photon state with partial distinguishability error ^′ may be distilled out of ^ = 5photons with error ^ as produced by imperfect sources.The figure further shows that the 5-mode Fourier transform has multiple –in this embodiment seven - herald patterns which may result in a reduced error. Inparticular, the figure shows a first graph 502 associated with a herald pattern (1,2,3,4), i.e.one detected photon at output ports 1-4, a second graph 504 associated with heraldpatterns (1,1,4,4) and (2,2,3,3) and a third graph 506 associated with herald patterns(1,1,1,2) (1,3,3,3),(2,2,2,4) (3,4,4,4). Each of these measurements corresponds – togetherwith a photon in mode 5, with an allowed outcome of the Fourier interferometer. Theseherald patterns can be computed in advance. All other measurement outcomes cannot be used for a purification scheme.The figure also shows that the purification process only works when the initial error ^ isbelow a herald measurement-dependent threshold, which may be different for differentherald patterns. Herald pattern (1,2,3,4) can only be used for photon distillation if the initialerror of a partial indistinguishable photon is approx. below 0.08 (8%). Similarly, the heraldpatterns (1,1,4,4) and (2,2,3,3) can only be used for photon distillation if the initial error ofthe photon mode is smaller than approx.0.42 (42%). For the third herald patterns theinitial error of the photon modes needs to smaller than approx.0.48 (48%).The data presented in Fig.5A shows that for certain ranges of initial errors,multiple herald patterns exist. In that case, whether the heralding scheme succeeds ornot, and the degree to which the partial distinguishability is reduced, will depend on the specific heralding pattern, the initial error, and the error threshold associated ^^^^associated with that heralding pattern For sufficiently low initial error, the data show thatthe new error scales as ^′ ≈ independent of the actual herald measurement.The heralding patterns, the thresholds and the reduced errors associated with each of theherald modes may be computed in advance for a certain transformation implemented onthe interferometer. These parameters may be stored in a database as described withreference to Fig.3 and used to control further circuitry.Fig. 5B depicts a plot of the probability as a function of the initial error. Theinformation in this plot can be used to calculate the expected herald probability outside the low error regime, which is needed to calculate the expected resource costs. Further, this information can be used to estimate the weighted error reduction and therefore isimportant when computing an (adaptive) distillation strategy.Fig. 6 depicts a method for photon distillation according to an embodiment.As shown in the figure, the process may start with a step of providing partiallyindistinguishable photon modes with an initial error to the input of a N-mode linearinterferometer which is configured to transform a product input state of the photon modesinto an entangled multimode state (step 602), preferably the linear interferometer beingconfigured to perform a Fourier transform or a Hadamard transform, the outputs of thelinear interferometer including N-1 ancillary outputs and a herald output. Then, measuringthe ancillary outputs of the linear interferometer and determining if the measured valuesmatch one of a plurality of herald patterns associated with the configured linearinterferometer (step 604); and, generating a herald signal indicative of a photon with anerror that is smaller than the initial error at the output mode of the linear interferometer ifthe measured ancillary modes match one of the plurality of herald patterns (step 606).Hence, the inventors found that for certain interferometer implementationsmultiple herald patterns exist, wherein each herald pattern is associated with a certainerror reduction of the initial error of partially indistinguishable quantum states of light thatare provided to the input of the interferometer. As is shown below in detail, theconsequence of the existence of multiple valid heralding patterns is that the number of quantum states of light required to achieve a constant error reduction is substantiallyreduced compared to the state of the art. Whereas previously, the quadratic scaling was^ required to achieve a constant fractional improvement in the indistinguishability, i.e. ^^ ∝^^, where ^ is the number of quantum states of light, e.g. photons, now linear scaling canbe achieved in the limit of asymptotically many quantum states of light, i.e.^ ^^∝ ^.The performance metrics as introduced in by Marshall, Distillation ofindistinguishable photons, Phys. Rev. Lett.129, 213601 (2022) consists of the number ofphotons required to achieve one purified photon of accuracy^^^ . As mentioned above, this metric is a combination of two factors: the probability that a valid herald pattern is observed, and the degree of improvement in partial distinguishability that is obtainedwhen a valid herald pattern is observed. From these two factors and elementarycombinatorial considerations, the resource requirements of a particular scheme may then be computed. The circuits found by Marshall provided a photon number cost that scaledquadratically as ^(( ^ ^^^) ). This scaling was computed based on required number ofphotons for a single distillation step and the gate success probability ^ (taken into accountthat for the three-photon distillation scheme ^ / ^ = 9, and the associated error reductionof ^^^= 3).To assess the quality of the distillation scheme, these two factors areconsidered. Below, it is shown that the error the error scaling ^′ ≈ holds truein general. Since the accuracy is ^^^≈ ^ independent of the herald measurement, allindividual success probabilities of herald measurements may be combined into one gatesuccess probability ^. Then, the photon number cost can be modelled as ^(( ^^^^) ), where It is noted that ^ is the marginal probability of observing exactly one photonin mode ^. All measurements contributing to this marginal probability are valid heraldmeasurements, i.e. the herald modes. The contributing measurements in the low errorregime are the measurements where the total output pattern, which consists of exactlyone single photon in mode N and where the ancillary N-1 photons are distributed over theherald modes, is not suppressed according to a zero-transmission law. The marginalprobability ^ may be computed by adapting the extended sample space formalism ofClifford & Clifford to find This equation may be evaluated numerically to find ^. As is shown in Fig.7 , wenumerically find that in the limit of large N, p tends to ¼. From this, ^ may be computed by evaluating Eq. 6, which has the limiting behavior ^ → 1 in the limit of large N, as claimed.FIG.7 depicts the error reduction photon-cost scaling for photon distillationschemes as described with reference to the embodiments in this disclosure. As shown inthis figure, in the limit of small partial distinguishability error, the required number ofphotons to distill one purified photon is ^(( ^^^)^). The scaling coefficient ^ is computed as afunction of the ^-photon Fourier transform (represented by black dots). The scaling of ^ ismade visible up to ^ = 50. The calculations suggest that ^ ≥^ ^ ^and ^^^^ →^ ^ = . Therefore, as a rule of thumb for the photon number scaling coefficient, Eq.8 only slightlyoverestimates the required number of photons for ^ ≥ 5. The approximation improves forincreasing N. The proposed rule of thumb (Eq.8, blue line) slightly overestimates theexact calculations. Only ^ = 3, 4 follow the quadratic scaling law as found by Marshall(dashed line). Fig. 8 depicts a photon distillation system according to anotherembodiment. The photon distillation system may comprise at least two concatenatedphoton distillation sub-systems, a first photon distillation sub-system 8021 and a secondphoton distillation sub-system 8022. Each of the photon distillation sub-systems mayinclude a photonic integrated circuit, a detection module and an optical switch similar tothose described with reference to the photon distillation system of Fig.3. The detectionmodules may be controlled by a computer 804 which is connected to a storage medium806 for storing errors. These errors may be stored during the photon distillation process tokeep track of the error of distilled photons. The computer is further connected to adatabase 808 which comprises information (characteristics about the distillation schemethat is implemented by the first and second distillation sub-system. The computer may be configured to execute a photon distillation process,wherein the first photon-distillation subsystem executes a first distillation process resultingin the signalling of a first herald pattern xxx to the computer. The computer may processthis information based on the characteristics of the first photon distillation sub-system (the herald patterns and the associated errors and error thresholds) that are stored in thedatabase. Based the signalled herald pattern, the computer may determine a distillationstrategy which may include the use of the second distillation sub-system The error ofheralded distilled photons may be stored in the storage medium so that this informationcan be used when further processing distilled photons. Further, the computer may control the routing of the distilled photons basedon the signalled herald pattern. For example, distilled photons 810 processed by the firstphoton distillation sub-system that meet target error specifications (e.g. an error below acertain target error) may be routed to an output of the distillation system so that thedistilled photons can be used in further quantum information processing. Distilled photons812, which do not meet the target error specifications (e.g. an error below a certain targeterror) may be forwarded to the second photon distillation sub-system (or discarded),which may use the distilled photons in a further distillation step to further reduce the error. Execution of the second photon distillation sub-system may result in thesignalling of a second herald pattern xxx to the computer, which may process thisinformation based on the characteristics of the second photon distillation sub-systemsorted in the database. If distilled photons produced by the second photon distillation sub- system meet the target error specifications, the distilled photons may be routed by thecomputer outside the distillation system for further quantum information processing.Fig. 9 depicts a photon distillation system according to yet anotherembodiment. The photon distillation system 900 may comprise a computer 902 configuredto control different photon distillation sub-systems 9041,2 of different sizes. For example, afirst photon distillation sub-system 9041 of a first size and second photon distillation sub-system 9042 of a second size may receive input photons 906 associated with a certainerror. Detection modules 9081,2may detect herald patterns of the ancillary photon modes and measured herald patterns or information about the herald patterns may becommunicated to the computer, which subsequently controls a switch 910 which may beprogrammed to control and / or route distilled photons, including for example: 1)discontinue a distilled photon; 2) group equal error photons for processing in a nextdistillation step 912; and / or 3) release distilled photons 914 that meet the target errorspecifications. Below the physics behind Fourier transform-based purification scheme isdescribed in greater detail. The schemes are most conveniently analyzed in the limit oflow partial distinguishability error ^. To that end, the input product state may be rewrittenas (Eq.1) The probability of measuring a particular mode assignment list ^^ is given by ^^(^) ∶= where ^^defines the ^-photon interference contribution

[0021] . This may be approximated by the expression: where ^^ ∶= ∑^ ^^^ ^^ is the measurement probability for fully indistinguishable photons. Inthe context of mode-assignment lists ^^, the conjecture ^^maybe postulated, which is supported by numerical evidence computed for ^ = 3 to 10. Incontrast, computations suggest ^^ ^ ≤ (^ − 1) ∑^ ^^^ ^^for Haar-random unitarymatrices. The conjecture may be used to write On the other hand, based on (Eq. 5) the measurement probability may be written aswhere λ^is the conditional probability of measuring outcome ^^given that one of thephotons is distinguishable. Solving for λ^, it is found that λ^ =^^^. The new error ^′ may bewritten as which confirms the hypothesized ^′ ≈ + ^(^^) scaling for Fourier transform-basedpurification methods. Finally, Bayes’ theorem states that the posterior measurement probability of indistinguishable photons is: where our conjecture states^^^ (^^) = ^^^^(^ ^^) > 1 for ^ > 0. This implies that all modeassignment lists ^^satisfying the complimentary zero transmission law, so all modeassignment lists that do not constitute a forbidden outcome by a zero-transmission law (asdefined by Tichy, Malte Christopher, et al. "Zero-transmission law for multiport beamsplitters." Physical review letters 104.22 (2010): 220405. are valid herald measurementsand therefore contribute to the gate success probability ^ = ∑^^ ^^ .Finally, we give an analytical argument why the success probability ^ of our heraldingscheme must approach ^ =^ In the limit of large N, it is reasonable to assume that the probability of a particular outcome will on average not be affected by whether thatoutcome has exactly one photon in mode ^ or not, which is the condition for photondistillation to occur. This means that we can obtain the success probability of our schemeby a simple counting argument: there are ^2^ − 1^^ measurement outcomes in general of^ which a fraction ^are allowed outcomes. There are ^2^ − ^− ^ measurements in which^ there is exactly 1 photon in output mode ^, of which also a fraction ^ are allowedoutcomes. Dividing these two gives a probability ^ =Then, it follows naturally that The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting. As used herein, the singular forms "a," "an," and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms "comprises" and / or "comprising," when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and / or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and / or groups thereof. The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present invention has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to the invention in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the invention. The embodiment was chosen and described in order to best explain the principles of the invention and the practical application, and to enable others of ordinary skill in the art to understand the invention for various embodiments.

Claims

CLAIMS 1. Method of reducing a partial distinguishability error of a quantum state of light comprising: providing partially indistinguishable quantum states of light with an initialerror to the input of at least one first N-mode linear interferometer, which is configured totransform the quantum states of light into an entangled multimode state, the N outputs ofthe first linear interferometer including N-1 ancillary outputs and a herald output;determining photon number measurement , the determining includingmeasuring quantum states of light at the ancillary outputs of the linear interferometer;determining if the measurement values match one of a plurality ofpredetermined first herald patterns associated with the configured first linearinterferometer, each of the first herald patterns being indicative of a state of the first linear interferometer in which, depending on the initial error, the first linear interferometer cangenerate a quantum state of light with a reduced partial distinguishability error at theherald output; and, generating a herald signal indicative of at least one quantum state of lightwith a reduced partial distinguishability error at the herald output of the first linearinterferometer if the measurements values match one of the plurality of first heraldpatterns and if the initial error is below a predetermined threshold error associated withthe matched first herald pattern.

2. Method according to claim 1 further comprising : controlling an optical switch at the herald output of the first linearinterferometer based on the herald signal.

3. Method according to claims 1 or 2 further comprising: if the reduced partial distinguishability error of the quantum state of light atthe herald output of the first linear interferometer is smaller than a target error, providingthe at least one quantum state of light with the reduced partial distinguishability error to asecond linear interferometer, which is configured to transform quantum states of light intoan entangled multimode state, which has ancillary outputs and a herald output, and which is associated with one or more predetermined second herald patterns, each of the second herald patterns being indicative of a state of the second linear interferometer in which,depending on the reduced partial distinguishability error of the quantum state of light, thesecond linear interferometer can generate a quantum state of light with a further reducedpartial distinguishability error at the herald output of the second linear interferometer.

4. Method according to any of claims 1-3 wherein the number of modes Nof the first linear interferometer is larger than 4.

5. Method according to any of claims 1-4 wherein the quantum states oflight are single photon states, multi-photon states, squeezed states or optical cat states.

6. Method according to any of claims 1-5 wherein the first linearinterferometer is configured based on a transformation matrix for performing an opticaltransformation comprising one or more forbidden outcomes and / or based on a transformation matrix for performing an optical transformation wherein the transformation matrix exhibits symmetry in at least part of the rows and / or columns of the transformation matrix.

7. Method according to according to claim 6 wherein the linearinterferometer is configured to perform a Fourier transform or a Hadamard transform.

8. Method according to any of claims 1-7 wherein the first linearinterferometer is implemented as programmable universal multiport interferometer.

9. Method according to any of claims 1-8 wherein the configuring of the firstlinear interferometer is based on a transformation matrix, the transformation matrix beingbased on a product of ^^,^matrices, a ^^,^matrix defining a lossless beam splitterbetween channel ^ and ^ with reflectively ^ and phase shift ^.

10. Method according to according to claim 9 wherein the reflectivity andthe phase shift parameters of the ^^,^ matrices are used to configure the first linearinterferometer.

11. A system for of reducing a partial distinguishability error of a quantumstate of light comprising:at least a first N-mode linear interferometer, which is configured totransform quantum states of light into an entangled multimode state, preferably the firstlinear interferometer being configured to perform a Fourier transform or a Hadamardtransform, the N outputs of the first linear interferometer including N-1 ancillary outputsand a herald output;an optical source module configured to generate partially indistinguishablequantum states of light with an initial error and providing the partially indistinguishablequantum states of light to the input of the first linear interferometer;a detector module for determining photon number measurement values ,the detector module being configured to measure the ancillary outputs of the first linearinterferometer; and, a processor configured to: determine if the measurement values match one of a plurality ofpredetermined first herald patterns associated with the configured first linearinterferometer, each of the first herald patterns being indicative of a state of the linear interferometer in which, depending on the initial error, the first linear interferometer cangenerate a quantum state of light with a reduced partial distinguishability error at theherald output; and, generate a herald signal indicative of at least one quantum state of lightwith a reduced partial distinguishability error at the herald output of the first linearinterferometer, if the measurements values match one of the plurality of the first heraldpatterns and if the initial error is below a predetermined threshold error associated withthe matched first herald pattern.

12. System according to claim 11 further comprising: an optical switch at the herald output of the linear interferometer;wherein the processor is further configured to control the optical switch at based on the herald signal.

13. System according to claims 11 or 12 further comprising: asecond linear interferometer, which is configured to transform quantumstates of light into an entangled multimode state, which has ancillary outputs and a heraldoutput, and which is associated with one or more predetermined second herald patterns, each of the second herald patterns being indicative of a state of the second linearinterferometer in which, depending on the reduced partial distinguishability error of thequantum state of light, the second linear interferometer can generate at least onequantum state of light with a further reduced partial distinguishability error at the heraldoutput of the second linear interferometer; wherein the processor is further configured to provide the at least onequantum state of light with the reduced partial distinguishability error to the input of thesecond linear interferometer, preferably if the reduced partial distinguishability error of thequantum state of light at the herald output mode of the first linear interferometer is smaller than a target error.

14. A computer program or suite of computer programs comprising at least one software code portion the software code portion, when run on a computer, being configured for executing the method steps according any of claims 1–10.

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