GB2620961A - Methods for using quantum computers - Google Patents

Abstract

A method that outputs a quantum algorithm and corresponding circuit ("quantum circuit P") that encodes a multivariate probability distribution (multivariate distribution) wherein the distribution contains dimensions that represent maxima and minima of two correlated random variables that correspond to other (new) dimensions generated from the distribution and wherein the quantum circuit uses a threshold condition and corresponding truth qubit to calculate upper and lower bounds of a random variable of one of the dimensions of the distribution and an additional (flag) qubit computed from an exclusive sum of products of a truth value of the truth qubit wherein the flag qubit acts as a conditional control (flag qubit set to value "1"). In this way, the further “flag” qubit may act as a conditional control in quantum Monte Carlo integration (QMCI) which may be beneficial in applications what use QMCI computations (e.g. financial and science data processing).

Description

METHODS FOR USING QUANTUM COMPUTERS

Technical Field

The present disclosure relates to methods for using quantum computers, in particular to methods for using an enhanced state preparation circuit builder for generating quantum circuits for execution on quantum computing hardware for summing and taking maxima/minima of correlated random variables and encoding threshold conditions in an amplitude of a qubit. Moreover, the present disclosure is also concerned with software products for execution on quantum computing hardware for implementing the aforesaid methods.

Background

It is known that quantum computers can be configured to perform quantum Monte Carlo integration (QMCI). In a Europe patent application EP22159844.4 and in a US patent application 17/684254, various methods of quantum Monte Carlo integration are described For many applications of such integration, it is necessary to compute advanced functions which have required computing tasks to be divided between a classical computer and a quantum computer, wherein the classical computer and the quantum computer are configured to function as a hybrid computing arrangement. In order to enhance computing performance, it is desirable to increase a proportion of computations to be performed being executed by the quantum computer. Such advanced functions can include integration as well as detection of maxima and minima, and computer whether or not one or more thresholds have been traversed.

Summary

According to a first aspect, there is provided a method for using a quantum computing apparatus to prepare a state therein that encodes a target probability distribution, wherein the method includes steps of: (i) receiving at the quantum computing apparatus first input data defining one or more dimensions (d) of the target probability distribution; (ii) receiving at the quantum computing apparatus second input data defining one or more qubits corresponding to the one or more dimensions (d), CO receiving at the quantum computing apparatus third input data defining a support for the one or more dimensions (d); (iv) using the quantum computing apparatus to generate from the first, second and third data a quantum circuit P that provides a representation of the target probability distribution, wherein generating the quantum circuit P includes at least one further steps of: (a) using the quantum computing apparatus to sum random variables in steps (i) to (iii) to generate a new dimension of multivariate distribution as being the quantum circuit P; (b) using the quantum computing apparatus to calculate for step (a) at least one of maxima and minima of two correlated random variables represented as dimensions of the multivariate distribution and to generate a new dimension containing the maximum / minimum accordingly; (c) using a threshold circuit of the quantum computing apparatus to calculate upper and lower bounds of a single random variable represented as one dimension of the multi van i ate distribution; (d) using the quantum computing apparatus to add at least one extra truth qubit to one or more qubits of the quantum circuit P to check if a threshold condition is satisfied in steps (b) and (c); (e) adding at least one further flag qubit, wherein the at least one further flag qubit is computed from an exclusive sum of products (ESOP) of at least one truth value of the at least one extra truth qubit, wherein the exclusive sum of products is computed from each term of the exclusive sum of products; and outputting the at least one further flag qubit when set to a value 1, and outputting the corresponding quantum circuit P. The invention is of advantage in there is provided a series of steps, according to user-defined instructions, that enhance a quantum state preparation circuit encoding some multivariate probability distribution such that the circuit further includes dimensions pertaining to sums and maxima / minima of other dimensions, and optionally includes a further "flag" qubit that will act as a conditional control in quantum Monte Carlo integration (QMCI). The series of steps are especially motivated for financial QMCI computations, but is expected to find application in many areas such a high energy physics computation, astrophysics computations (for example, processing satellite-collated images) and so forth..

Optionally, in the method, the quantum circuit P is configured for implementing quantum Monte Carlo integration (QMCI).

Optionally, in the method, there is used a plurality of the dimensions (d).

Opt onally in the method, the summing in step (a) is implemented using a binary adder circuit.

According to a second aspect, there is provided a quantum circuit P that is generated using the method of the first aspect.

According to a third aspect, there is provided a quantum computing apparatus that is configured to prepare a state therein that encodes a target probability distribution, wherein the quantum computing apparatus is configured in use: (i) to receive at the quantum computing apparatus first input data defining one or more dimensions (c/) of the target probability distribution; (ii) to receive at the quantum computing apparatus second input data defining one or more qubits corresponding to the one or more dimensions (d), (iii) to receive at the quantum computing apparatus third input data defining a support for the one or more dimensions (d); (iv) to use the quantum computing apparatus to generate from the first, second and third data a quantum circuit P that provides a representation of the target probability distribution; wherein the quantum computing apparatus is further configured to generate the quantum circuit P by: (a) using the quantum computing apparatus to sum random variables in (i) to (iii) to generate a new dimension of multivariate distribution as being the quantum circuit P; (b) using the quantum computing apparatus to calculate for (a) at least one of maxima and minima of two correlated random variables represented as dimensions of the multivariate distribution and to generate a new dimension of containing the maximum / minimum accordingly; (c) using a threshold circuit of the quantum computing apparatus to calculate upper and lower bounds of a single random variable represented as one dimension of the multi van ate distribution; (d) using the quantum computing apparatus to add at least one extra truth qubit to one or more qubits of the quantum circuit P to check if a threshold condition is satisfied in (b) and (c); (e) adding at least one further flag qubit, wherein the at least one further flag qubit is computed from an exclusive sum of products (ESOP) of at least one truth value of the at least one extra truth qubit, wherein the exclusive sum of products is computed from each term of the exclusive sum of products; and outputting the at least one further flag qubit when set to a value "1", and outputting the corresponding quantum circuit P. Optionally, in the quantum computing apparatus, the quantum circuit P is configured for implementing quantum Monte Carlo integration (QMCI).

Optionally, in the quantum computing apparatus, the quantum computing apparatus is configured to use a plurality of the dimensions (d) Optionally, in the quantum computing apparatus, the quantum computing apparatus includes a binary adder circuit for implementing summing in (a).

According to a fourth aspect, there is provided a software product that is executable upon the quantum computing apparatus of the third aspect, to implement the method of the first aspect.

Additional aspects, advantages, features and objects of the present disclosure would be made apparent from the drawings and the detailed description of the illustrative embodiments construed in conjunction with the appended claims that follow.

It will be appreciated that features of the present disclosure are susceptible to being combined in various combinations without departing from the scope of the present disclosure as defined by the appended claims.

Description of diagrams

Embodiments of the disclosure will be described with reference to the following diagrams: FIGs. 1 to 16 are depictions of quantum circuits that are executable on quantum computing hardware (or a quantum computing hardware emulator) for implementing

embodiments of the present disclosure.

In the accompanying diagrams, an underlined number is employed to represent an item over which the underlined number is positioned or an item to which the underlined number is adjacent. When a number is non-underlined and accompanied by an associated arrow, the non-underlined number is used to identify a general item at which the arrow is pointing.

Detailed description of embodiments

1 Introduction and Motivation

Every quantum state can he interpreted as an encoding of a probability distribution -most notably when said state is measured in the computational basis, and this a given bitstring is sampled with probability equal to the squared amplitude of the corresponding term when the state is expressed as a superposition of computational basis states. Preparing states that deliberately encode particular target distributions is an important task in many quantum algorithms, in particular in quantum Monte Carlo integration (C2MC1) RA.

We use 'P' to denote a circuit that prepares a state, 1,0, that is Ip) = P (where Ill signifies an appropriate number of qubits in the zero state), which is interpreted as an encoding of a multivariate probability distribution ip)= E 1/4( ),...,x(d))1a,(1) ro) (1) 11e jN1 r(dlc fa, (i'd In order for such a prepared state to be correctly interpreted as an encoding of a probability distribution, we can see that it is necessary to supply the following additional information: 1. The number of dimensions of the probability distribution, denoted 4 (univariate, 4 = 1, is a special ease).

2 The number of qubits corresponding to each dimension, denoted Ni N2, Ara.

3 The support of each dimension. For the dimension, x is a real number that denotes the minimum supported value; and AI° is a real number that denotes the spacing between points of probability mass.

Formally, in order to specify a state preparation circuit, as well as the circuit itself it is necessary to specify 'mete-data' describing how the state prepared by P is to he interpreted as a probability d'stributirm. This meta-data consists of a,x1, N which are respectively 4-length vectors of AN, 49, d is therefore implicitly defined as the length of these other inputted terms.

The circuit P is* typically treated as an input in QMCI, however it has been shown that it is* always pc-issible to construct a suitable circuit from the cottesponding etas:deal sampling, algorithm pj. We therefore procrerl assuming that a silitF111111, circuit P has 114.cri coil ti and the seCompanying meta-data is also inclitcled as an input. :bra the sake of definiteness.. in the following exposition we use the example of twat:tom pr oteSSeS used ni lmarii e ho eve, it will be appleCiated that the fiat:wing applies to any tither appliciation cii Monte Carlo integration. In many ithancial applications Of Monte Ciarlo integration, the various dimensions of the multivariate distribution pertain to financial time-serest datstaliced IM at (typically regular) intertials. The financial time-series data is the price (or the logagithrmof the price) of some underlying and in general many cod:elated.underlyings may be included in the same titultiaariate clistribu,tion teach Lime-series sliced at he -same nil cue Suitable niodels for SUCh MUIthuriate dietributions incliale multiymriate normals and lognormale, as welt as stochastic, time-series models suet as autoregressive All I models, samegiressive naiving-aVinage fARMA) modela hidden Mai kin. umilels and iii aehine learning genertathe models trained on historic time-sericis data, amongst others, NI-onto Carlo inregration. is then used to 11I1InrTically reliproximate ti.m. price of con tiC,I1Vnt:Rre instrn-meMs which depend on the underlyings, and various measures of risk pertaining to 'portfolios' containing the variemS finandal instruments (pcaeutially including derivatives). Typically, such Monte.Cario integrals are not siniple expectations Of the rendcan aMiables, out rather depend on certain threshold conditions. Furthermore, the expectation may be of the slam. average:or max / win of softie nutober of randbm variables pertaining to cations dimensions of the nnilliyariate larobabiiity distribution (note that average-is essentially lie alike ab UM for our purptises, as the slim will differ wily lw a -knoru n. c.onstant lactor Irma the. averige and this cepstant fat La ea:n.easly he accounted for classically) In order to enable any logical ameidoll cu the values (ded: is due-or false.) of the Vali UUS II teahold conditions, an optional further citibit is. added to act as a condition on any expectation value Computed.in QMCI, and this meana we must further specify in the information that acconipanies 4. Whether this optional tinbitis present, Thus we extend the meta-data to include 'flag-, which-is a 'booll that is one when the optional quint is present, and false otherwise. There is. also in general, the possibility that P may now contain further qubits thdt -are neither the Eag quint not pertain to the register Of the MultiVariate distributiOrk however in Our ClIrrel 1, embodiment, the explicibtlerhumicia climensioTIS (and their sizes) and of wirether there is a flag quint Means that tins inibrmation can lac left. hiaplicit, In tins praper we propose a method that takes some -input (Moult F, along -with its original meta-data and flag: =U and rethres some enhanced wirsicin P (with possible further diniensions. corresponding to max / min and:puns of the dimemaolts, an specified by the user:), along with an as updated iiet at metaselatal and may have flag.= 0 or.flag =I.

This additional qui* :simply acts as a control cubit en the.Q11.1C.I circuitry, and ao it is simple to incorporate this additional degree of freedom into any QMCI. In the case of Fourier CALM Ill, the circuits generated would be of the form: as illustrated in FIG. 1, where the 5ig.,-,0" quints are the extra. Titbits that are neither those pertairnug to the dimensions of the multivariate probability distribution, nor the nag qubit.

Such an enhanced P is therefore manually constructed to allow the user to programme any financial derivative / risk calculation where Lite quantity to be calculated depends on: 2:1:11th and maxima / minima of the correlated random variables; as well as optionally conditioned en a. further miltit, which is itself a function of the values (true of false) cif one or more thresholds. This concept is developed and detailed in full in the next section, however for now we note that MAYA of the common financial derivatives are covered by this framework: Ion C8iaMurny Ifs leis -pne applied f is!LtIons preIerr&d Loh price s At ease timostep, Log Price space duperdhg on opt.mal strategy Like Arnericen with tog price space discrete time At eachtime step r oat:On r,:irrt[21 Uaskets of options Con he a basked of Price space army.o the other t.ioos hare Asian options Average et, thecae option na OnL3ass6ttvatue tither If Chooser option ea One (or moret to decide whether call r put og Pit ace took back options Max J Min Like a European tog price space opt on on the max/ ( -tavola} f (x) estaltd ock in!knock American POW, oradden option cas option on the average Pit, space can included n its own dght b w dentrathe yeti May Construct.

could have binary version of or a basket of opttons, the weghts of the various items h the basket may not be equal, Thu order-ti amply sum bit ironic; we actuairynoca to acaust the O tor cacti dim:intim such that he bitstr rig sr anolic try cludes the correct weehis lhe method in,r achieving this presented het:ein pi, s Instnettans on the oneadLag et the strip tt et Jte v:31 ions tianea,,nan U ILO *iii_ liYIflP7e ii)11 ii Reble blatit±. C.4iFX#1113X+ prelf evilly thC Ti'VrIt,11111' xi ms rht'Ten'). This Iflef YRS titat the following sliotild he taken not catty a series of steps for building a circuit, but also as a procedure for checking and cl consistently

-

updating the meta-aata escrilaing P. suck' that the returned (enilancest) P Is always wilici and correct a6 or so: Prepare a new (01 1, dimension of the mutt:ye:late ttribution such that.

4.64.1; = 401 = Technical Detail and Steps of the Method ABumaiddui module haat rnotnkesreassinput a circuit of type without a flag qubits, and returns a pe P, with or without a flag cubit.

sophisticated circuit P orcuit The input circuit will, in general, have some d dimensions. As per the in?concerning how to interpret Pr. each dimension will be assigned a number ofwires, d, arx(ra d. A st the a"random variable be denoted fr-tn, so we have We then break down the construction into four parts.

Summing random variables First we need a programming environment to allow the use to (reversibly surnY random variables such that the summed random variable appears as a new dimension of the multivariate distnbution.

An instruction such as: (where i and i are distinct integers betwben land d Ceeck 3Y13 = it not then output r3n error me mere) thus require he foli inp. steps; in and stop, maxi (the number of wires for Vile then trust append to P a reversible circuit to sum Id put the result as

E

cone usirg a reversible binary adder circuit. Sirially, we set fl 3% d* Any number of such sums are Mi to be performe including on di skins that were orsated ' previous sums.

Taking Meath= (minima We alto permit the (4E1' tO create Flew dimeilsiOne of theMUltivatiete distributiten corresponding to the maximum / minIMUM of two existing dimensions.

An instruction such as: Cr (where land] are distinct Integers between and t1 lncktslue) thus requires the foili * Cheek xr -= 46') = t5i(° FR not then output an error message and etop.

Otherwise prepare a newit" li)th dimension of the rnuttiveriate distribution such that *(d4-1). e, Are * ^ Aid; reardlein hid We then must append to Pe reversible circuit (which contains a control qubit with an implementation dependent final ealue) to tate the maximum / minimum of ew and icul and put the answer as T'S'i. This can be achieved using similar circuitry as for the threshold (see below;.

* Finally we set ft-d+ 1 Thresholds The next step involves writing circuitry to compare the val thresholds. The user will include instructions of the form' ties various random variables Threshele(i, value, terre) Which means that the random variable in the tth (emersion will be compared to value, 'type has a value of either 'BoundTypelower or 'Bound li ypeelpper' -if its 'Boundlypelowee, this means the threshold is tisfied if the random variable exceeds value; and if d is 'BoureiTypeOppee it means the threshold is atisf led if value is equal or exceeds the random variable, We now do the following: * Add a new set of IV, -1 1 qubits to take the result of the subtraction of the value from era. Since these yen quiets do not belong to a dimension, there needs no xi and S be associated.

* Reversibly check the value of euij against value by subtracting the latter from the former, in the case of 'BoundType.Lowee, the most significant qubit will be set to one if the threshold condition is satisfied. For a 113ouneType.Uppers condition, this qubit needs to be inverted.

This can be repeated any number of limes, but it is necessary to keep track of the qubits that have been added as 'threshold qubits' for the final step, below.

Preparation of the flag qubit 1. Append one further qubit (the 'flag' qubit); 2. The user should input a logical expression as an exclusive um of products (ESOP) of the various threshold truth values prepared in the previous step.

3. Each product is then a multi-controlled not the terms in the product as the conditions, and targeting the qubit added in step one.

4. Doing the previous step for each term in the ESOP (each time targeting the same qubit) automatically achieves the exclusive sum.

Note that the order of sum; max/min and threshold is Important Maximum, minimum a etationt oar) be called in any Order, but Mate must all precede all of the threshold ope The final user input is the logical expression depending on the threshold terms Reversible circuits for the sum, main and threshold operations Here we give simple reversible implementations for first binary sum, and then threshold.

Reversible sum A circuit to take two registers, and 'c of sizes; and Nr respectively (where we let f simplicity) and outputs their sum, denoted X (which has N.i M I bits).

To do this, we can use Toffoli and CNOT gates tc perform 'in-place' reversible sums. Let xt" and xul be two numbers to be added (not necessarily using the same number of bits when represented in binary) to give -'444b. The essential circuit for the case PI, k is then as illustrated in FIG. 2.

Once N1 -k NI, the corresponding wire xi, which only ever acts as control) and any gates operating on this wire vvill be omitted resulting In a quantum circuit as depicted in FIG. 3.

Reversible maxi min The conditions mat(e, x(iii) and ininfx ii'D) lead to the appropriate one of the compared random variables being copied into a newly created dimension of the multivariate distribution. in order or the condition to be encoded in a summation circuit, one of the two values has to be inverted with X gates. If the dimensions have an unequal number of qubits (without loss of generality we can assume Nd0.-5 Ni as the dimensions can be switched accordingly), then the dimension with more qubits is inverted, so as to not lose the inversion of leading qubits.

After the uncornputation of the summation, there will be fWd.t free qubits which will take the greaterfiesser of the two random variables as a result, depending on the control qubit e depicted in FIG. 5, for NI d+] = N1 = NJ and depicted in FIG. 6, it °Poi depicted in FIG. 4, where the result is x(d = min(x and the value of the control qubit is = 1 if A1 < ancl c 0 klisK"> -'4')Note that the summation circuit can be trivially inverted by versing the order of the gates.

The summation circuit and its inverse are the same as for the summation opera ion. The circuits PRE for preserving the control quialt are !he COPY circuitfor al&htitig the minimum is as depicted in FIG. 7, or the case N. = Mi. For Ni> A theToffoli gates connected to non-e calculating the maximum, the order ci copying xi and -T is switched.

Reversible threshold The threshold circuit is similar to the circuits for calculation of the maximum Cr minimum of two random variables. The notable difference is that the second value is a fixed classical value and therefore not encoded in a quantum register.

It is convenient to define the threshold value in the fallowing way: * If the user specifies a lower-bound, this is taken as a strict inequality. The output will only be set to lit the value in the register strictly exceeds the threshold, and olhervvise (if less than or equal) the output is O. * If the user specifies an upper-bound, this is taken as an inequality that can be saturated, that is if the value in the register is less than or equal to the threshold then the output will be set to one; it will only be set to zero in the case that the upper-bound is strictly exceeded.

-12 -The idea is to use the abovesummation blecka to determine it a soecified threshold is breached. Newsy since the threshold is a classical number, it d n not mad to be put into a trig * Let the number beino cornparee to the threshold be, a N. -bit number, * We threshed is inputted as a pure number, let this be T, and the first step is to convert this into a binary string, f, according to the xi. and A for the register in questton. This binary string should be obtained by rounding T down, $o when the b'diety sting, t, c interpreted as tvciut T xl A according to -Xikind A, then this value should be less than or equal to Concretely, = Friand consequently 5-F < 1' A..

* We now obtain f' as the one's complement of t according th: r rriOte, that he NJ a binary string et N, °nest.. rft is obtained classically. * Pis itself a binary string of N, bits.

We now sum v and t' to give an N, bit number, r. f he most significant hit c ill exactly if t'n tt. f ef x"1/4 ' A this is egiavalent to = Or A Note that with these conventions, tire value t= 2' -1 <t>. P = Ci yields a trivial condition, as the lower newel is never satisfied, whereas the upper bound always is Useful threshold values are therefore T, yielding e O. 1, * * To achieve the threshold calculation, we can e surrey, II logic above but without putthig inf register. The full circuit is given by by a quantum circuit as depicted in FIG. 8, where the value of c is c = I if xl.t1 > r sad e = 0 if at' . For an upper bound, the resulting threshold control habit c is Inverted with an X gate.

The circuits for adding an implicit binary digit are given by a quantum circuit as depicted in FIG. 9, for 4 ° and given by a quantum circuit as depicted in FIG. 10, for 4. = 1.

The most significant qubit of the result is the threshold control qubit. The other result qubits are "uncomputed" for reuse as ancillas. The uncomputation circuit is the reverse (adjoing of the summation circuit, with a preservation circuit before that prevents the uncomputalion of the threshold control qubit. The preservation circuits are similar to their counterparts in the Min-Max calculation, but with the second register omitted, since the threshold value is given implicitly. They are given by a quantum circuit as depicted in FIG. 11, -13 -for 4 -11 and a quantum circuit as depicted in FIG. 12, for 4 tr 1. Here C reret denotes the threshold control qubit.

Note that the total number of wales needed Is the maximum number of calcifies over all threshold blocks, since they can be reused.

Planned optimisations * lithe thresholds are ordered in decreasing number of qubits then ancillas may be converted into controls of following thresholds.

* Controlled operations at the beginning of a circuit can be removed if the incoming qubits are known to be in the 0 state.

* The conservation circuit can be merged with the adjoined summation circuit.

* If the same threshold value is used both as a lower/ and upper bound, then only one such threshold circuit is needed.

a In the threshold circuit the value -"Q) can be inverted with X gates, instead of the threshold value t. This might lead to a lower usage of X gates, depending on the threshold value and possibly compiler opfinesation.

For the maximum/minimum circuit the dimension with fewer qubits can be inverted, and the leading is can be summed implicitly as in the threshold circuit. This might result in a lower number of X gates used, again also dependent on compiler optimisation.

a In a threshold circuit only as many qubits as are needed as amines/controls/flag for the following threshold circuits need to be uncomputed. The summation in the last threshold circuit does not need to be uncomputed, since the ancillas are not used again.

* One of the ancillas of the threshold circuit can be reused as the flag qubit of the ESOP circuit.

* Combining the above two ideas, just a single qubit of the summation in the last threshold circuit can be uncomputed (while leaving the others) to be reused as the flag qubit of the ESOP circuit.

3 Example

This example showcases the above: The Input ia a circuit P = J1 (that is simply 5 Hadamard gates on 5 qubits), which is to be interpreted as a 3 dimensional probability distribution such that * xm,xci have 2 qubits, o xelhast quhit.

o Far all, X, = Oz = 1 * Then we implement the following: o Create a new dimension: -ry'"' = O Create a new dimension: x = max (t o Threshold 1: x" O Threshold 2: xrm> 2 o Flag true if: Threshold I xor Threshold 2 Mate they cannot both we me so xor has the same effect as or here) -15 -Input and operatic) ender lying th dfatertsionat mottivoriate e cult = Circuit adonard gat-es provide tor a r.ffnn di 13 ci.tetrit..quIttits; e.111qubit diXt citation = Oat (iPutinfiCitcVit circuit =circuit triresui 2. 2, '11, 4t The fluster ot qubi Ls -(1,S, AP tliaen4ions stare II At/ crirstutsfpos store the silt* ri ion is a quthAt ci o.n c elf520 2 and by cot ro Zetin-, he resekr in d.toonsion 5.

deltaic:kr, 4. 5 los 'a dimensions I and 4, operations tSu2 3.1, Sae( 1, 411 he two thresholds are defined re Tap C.She ds -f disteosiatiu, vatueul, 1st, oral I tr hreshaldidisteesthnmS, vattl2, t-t.;:r4tostalf-TYYPPee.tto-rtsePerr.", ruhe,diudoett4inere011 ttoyer mut the Index-'llen:srPefevirsTretoll-)ilsecond. by.

ye num of two thrstithOd compt it co ataees art' so (isrled if the respeCeive -ChresAster Is Op

LC

it Inn to aas ore ed distrIO4t4de contai 01 oetion on an /Astarte anceisont porno ter'Sr enha d ci ttit_entiancer.enhanc istrievtiontrOistribution, ape ratiorts-pporations, h esholdsrifiresholds, es op.

eiCe circa tad Trol La55 on tt 6 a 1* a.

Enhanced circuit An enhanced circuit is depicted in FIG. 13, wherein: 1. The initial uniform distribution of the first 3 dimensions is prepared.

2. The random variables An' and X(3) are added Into dimension a S. As the first step of calculating the maximum of Tth and xth, the latter is awned.

4. The random variables xth and the inverted x441 are added into dimension s, with COtated0 taking the information which one is greater.

-16 -and wherein the enhanced circuit is further depicted in FIG. 14, wherein: evicus sun1mtbrciS t,nconcutea Ineersbn of A-14j is undone, and the g dimension 5.

eonnel quIrtit unaffected.

I s (according tc the control pubis) is e and wherein the enhanced circuit is further depicted in FIG. 15, wherein: 7. The first threshold is computed. with "nal taking the result. The enhanced circuit is depicted in FIG. 16, wherein: 8. The second threshold Is computed, with collft12 taking the result.

9. The ESOP of c"ir°11 and anr°12 is computed, with the flag qubif taking the result.

Output state The output state is an e4ual superposition of 32 eomputatonat basis states, each represented as one line output The fir'st tuple of each line contains the values of the 5 final dimensions where the first 2 are the input the fourth is the result of the summation, and the fifth is the result of the maximum calculation I he following triple contains the 3 control quaffs, the next number is the value of the 3 amilla outfits, wh,ch is always O. The final number is the gag crubl. Note that tie values at the control cubits are inconsequential far the Morita Carlo Integration. The flag qulnit indicates whether the 155OPthreshold ondition has been met References s. Herbert, "Quantum monte-carlo integration: The full advantage in inirnirnai circuit depth," 2021. [Online]. Available: Ittps://arxiv.orgfabs/2105.09100 [2] A. Montanaro, "Quantum speedup oF monte carlo methods," Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 471, no. 2181, p. 20150301, 2015. [Online]. Available: littpeproyalsocietypublisliing.orgldoijabs/10.1098/rspa.201.5.0301 [3] S. Herbert, "Every classical sampling circuit is a quantum sampling circuit," 2021. [Online]. Available: https://arxiv.org abs/2109.04842 1) -17 -7), 1, 2), 2. 2), 2, 211 3, 31, 3, 3), 4, 41, 0, 31, 2, 31, 2, a), 3. 31' 3' 31' 9, 4), 2. 0, 1, 0, 1, as 2' a, 0, a, 0), a a 0. 1) 0) a, e, a. 1) 1) -18-

Claims (10)

1. CLAIMS1 A method for using a quantum computing apparatus to prepare a state therein that encodes a target probability distribution, wherein the method includes steps of: (i) receiving at the quantum computing apparatus first input data defining one or more dimensions (a) of the target probability distribution; (ii) receiving at the quantum computing apparatus second input data defining one or more qubits corresponding to the one or more dimensions (a); (iii) receiving at the quantum computing apparatus third input data defining a support for the one or more dimensions (M; (iv) using the quantum computing apparatus to generate from the first, second and third data a quantum circuit P that provides a representation of the target probability distribution; wherein generating the quantum circuit P includes at least one further steps of: (a) using the quantum computing apparatus to sum random variables in steps (i) to (iii) to generate a new dimension of multivariate distribution as being the quantum circuit P; (b) using the quantum computing apparatus to calculate for step (a) at least one of maxima and minima of two correlated random variables represented as dimensions of the multivariate distribution and to generate a new dimension containing the maximum / minimum accordingly; (c) using a threshold circuit of the quantum computing apparatus to calculate upper and lower bounds of a single random variable represented as one dimension of the multivariate distribution; (d) using the quantum computing apparatus to add at least one extra truth qubit to one or more qubits of the quantum circuit P to check if a threshold condition is satisfied in steps (b) and (c); (e) adding at least one further flag qubit, wherein the at least one further flag qubit is computed from an exclusive sum of products (ESOP) of at least one truth value of the at least one extra truth qubit, wherein the exclusive sum of products is computed from each term of the exclusive sum of products; and (0 outputting the at least one further flag qubit when set to a value -1", and outputting the corresponding quantum circuit P. -19 - 2.
2. The method of claim 1, wherein the quantum circuit P is configured for implementing quantum Monte Carlo integration (QMCI).
3. The method of claim 1 or 2, wherein there is used a plurality of the dimensions (d).
4. 4. The method of claim 1, 2 or 3, wherein the summing in step (a) is implemented using a binary adder circuit.
5. A quantum circuit P that is generated using the method of any one of claims 1 to 4.
6. 6. A quantum computing apparatus that is configured to prepare a state therein that encodes a target probability distribution, wherein the quantum computing apparatus is configured in use: (i) to receive at the quantum computing apparatus first input data defining one or more dimensions (c/) of the target probability distribution; (ii) to receive at the quantum computing apparatus second input data defining one or more qubits corresponding to the one or more dimensions (d); (iii) to receive at the quantum computing apparatus third input data defining a support for the one or more dimensions (d); (iv) to use the quantum computing apparatus to generate from the first, second and third data a quantum circuit P that provides a representation of the target probability distribution; wherein the quantum computing apparatus is further configured to generate the quantum circuit P by: (a) using the quantum computing apparatus to sum random variables in (i) to (iii) to generate a new dimension of multivariate distribution as being the quantum circuit P; (b) using the quantum computing apparatus to calculate for (a) at least one of maxima and minima of two correlated random variables represented as dimensions of the multivariate distribution and to generate a new dimension of containing the maximum / minimum accordingly; -20 - (c) using a threshold circuit of the quantum computing apparatus to calculate upper and lower bounds of a single random variable represented as one dimension of the multi van ate distribution; (d) using the quantum computing apparatus to add at least one extra truth qubit to one or more qubits of the quantum circuit P to check if a threshold condition is satisfied in (b) and (c); (e) adding at least one further flag qubit, wherein the at least one further flag qubit is computed from an exclusive sum of products (ESOP) of at least one truth value of the at least one extra truth qubit, wherein the exclusive sum of products is computed from each term of the exclusive sum of products; and outputting the at least one further flag qubit when set to a value "1", and outputting the corresponding quantum circuit P.
7. 7. The quantum computing apparatus of claim 6, wherein the quantum circuit P is configured for implementing quantum Monte Carlo integration (QMCI)
8. 8. The quantum computing apparatus of claim 6 or 7, wherein the quantum computing apparatus is configured to use a plurality of the dimensions (d).
9. 9. The quantum computing apparatus of claim 6, 7 or 8, wherein the quantum computing apparatus includes a binary adder circuit for implementing summing in (a).
10. 10. A software product that is executable upon the quantum computing apparatus of any one of claims 6 to 9, to implement the method of any one of claims 1 to 4.