A method and apparatus for performing a projection of a quantum state onto a subspace

Abstract

An apparatus and a method are disclosed herein for performing a projection of a quantum state onto a subspace spanned by quantum states representing hypercliques which exclude a given set of hyperedges H comprising sets H2,... HD, where Hi contains hyperedges of size i. The apparatus is configured to use a classical computing system to find, for each 2=< i=< D, a set of hypercliques Ci of size >= i in the hypergraph whose hyperedges are the downward closure of those in Hi. The apparatus is further configured to construct a quantum circuit on first and second registers. For each i and each hyperclique K in Ci, a projection is performed on qubits of the first register onto the subspace spanned by quantum states, with a count of the overlap with K less than or equal to i being stored in the second register. These projections in series achieve the desired projection onto the subspace of hypercliques which exclude the hyperedges H.

Description

[0001] QTDA 2 - QUK226739-WO; MJJ / 65238 PCTA METHOD AND APPARATUS FOR PERFORMING A PROJECTION OF A QUANTUM STATE ONTO A SUBSPACE 5 Field The present applica^on relates to a method and apparatus for preparing a quantumstate represen^ng cliques of a graph. Background10 A network (graph) is formed of (i) nodes and (ii) edges between the nodes. Such agraph provides a rela^onal structure which is used to represent many forms of data, forexample, online social interac^ons, co-occurrence of biomarkers, geometric proximity and Bayesian nets. In some networks, edges may be directed from one node to another node. In the present context, the edges are undirected, i.e. an edge connects two nodes, but without 15 indica^ng any direc^onality between the two nodes. The cliques of a network are collec^ons (subsets) of nodes which are pairwise allconnected by edges. For example, assume we have a set of nodes n1, n2, n3 … n9, and anedge such as n2-n4 denotes a direct (without intermediary) link between node n2 and node n4.If the network includes edges such as n2-n4, n4-n7 and n2-n7, the nodes n2, n4 and n7 form a20 clique (n2, n4, n7). Such cliques o^en represent important features of the data and theinterac^ons between cliques may be used to capture geometric and topological informa^onabout the underlying network. Many graph algorithms involve manipula^ng cliques. A commonly used transforma^on on a graph is the construc^on of a meta-graph called the clique (junc^on) tree 25 whose nodes correspond to cliques and edges to connec^ons between cliques. For example, in the above example, the network might also have a clique (n1, n5, n8, n9) and there may be an edge n2-n8 which connects the clique (n2, n4, n7) with the clique (n1, n5, n8, n9). Such a meta-graph has a wide range of applica^ons, including for Belief-Propaga^on on Bayesian netsvia the Hugin algorithm (see, for exampleQTDA 2 - QUK226739-WO; MJJ / 65238 PCTh^ps: / / en.wikipedia.org / wiki / Junc^on_tree_algorithm), and more generally for many dynamic programs on graphs in which an algorithm is wri^en in terms of the clique tree. Some exis^ng work has involved construc^ng simplicial complex states (see, forexample, h^ps: / / en.wikipedia.org / wiki / Simplicial_complex). Such construc^on can be seen as 5 a general uniform quantum state prepara^on problem. This problem is generally thought to be difficult, requiring exponen^ally deep quantum circuits in general to prepare the uniform state over a chosen subset of the 2^ basis states of the Hilbert space on ^ qubits. Exis^ngmethods for tackling this problem include the use of classical op^misa^on rou^nes based on the Boolean sa^sfactory (SAT) problem (see, for example,10 h^ps: / / en.wikipedia.org / wiki / Boolean_sa^sfiability_problem) to compile quantum circuits for construc^ng these states. An example of the Boolean method is provided by Mozafari,Soeken, Riener and De Micheli

[0011] for preparing uniform quantum states.Quantum Topological Data Analysis (QTDA) is a quantum machine learning / data analysispipeline that involves manipula^ng quantum states which represent the simplices of some15 simplicial complex. Technical applica^ons of QTDA are iden^fied in US20240028939A1.Other uses of clique state projec^on and par^cular technical applica^ons can be found in US20240022247A1. Exis^ng work includes methods which are specialised for construc^ng simplicial complexstates. An example of this work for use with Vietoris-Rips complexes (see, for example,20 h^ps: / / en.wikipedia.org / wiki / Vietoris-Rips_complex) is the process described in Akhalwaya etal. [2], which provides a probabilis^c method for construc^ng the state from the graph ^.This method (process) in general produces smaller circuits than the general approach above. However, the quantum circuit produced by the method only succeeds with afixed probabilitywhich decreases rapidly for graphs ^ with few edges. This limits the prac^cal applicability of25 this method to complexes which are sufficiently dense in terms of the number of edges(compared to the total number of edges which are possible for a graph with a given number of nodes). The interac^ons between cliques can therefore be used to capture geometric and topological informa^on about an underlying network. One prac^cal problem with this use ofQTDA 2 - QUK226739-WO; MJJ / 65238 PCTclique informa^on is the exponen^al growth in the number of poten^al cliques in a network. For a network on n nodes the number of poten^al cliques of size k is {n choose k}, i.e. O(n^k). The present applica^on discloses, for a wide range of networks on n nodes, how toprepare and project onto quantum states on n qubits which store the cliques of the network in 5 superposi^on. Measuring and manipula^ng these states with other quantum circuits enables network analyses which extract topological and geometric informa^on about cliques from the network. Figure 1 provides an overview of the method described by Akhalwaya et al. [2] for construc^ng the quantum state encoding a Vietoris-Rips complex. The components of Figure10 1 may be regarded as opera^ons in the method and / or corresponding func^onal components(e.g. circuits) as appropriate. The top three boxes [red] 110, 120 and 130 describe the ini^alclassical processing of the graph data into sets of disjoint edges in ^^(two edges are edge- disjoint if they do not have any ver^ces in common). The two lower boxes marked with circlesin their corners [blue] 140 and 150 describe the opera^on of the quantum circuit whichfirst15 prepares a complete uniform state and then applies a sequence of quantum opera^ons and measurements based on the sets of edges determined classically (this sequence is described in more detail below with reference to Figure 3). On success of the projec^on circuits 150, thestate created is ^Γ^^. The process described in Figure 1 is based on a simple component circuit called an edge20 projec^on. This circuit is implemented on two registers, one with a qubit for each vertex of Γ(a set of simplices as defined in more detail below). The other register is for auxiliary qubits and comprises a single Toffoli gate (CCX gate) (see, for example, h^ps: / / en.wikipedia.org / wiki / Toffoli_gate) and a measurement. For any edge ^ = {^^ , ^^} inΓ, the corresponding edge projec^on has a Toffoli with controls connected to the qubits for 25 and ^^ and the target qubit a^ached to any free auxiliary qubit. This single Toffoli gateentangles any state on thefirst register with the target qubit on theauxiliary register to create the state |0^|^¬^^ + |1^|^^^ where |^^^ = ^^|^^ and^⊂^ ^^|^^. This single Toffoli gate is followed by a measurement of the targeted ^⊂^QTDA 2 - QUK226739-WO; MJJ / 65238 PCTauxiliary qubit in the computa^onal basis which has outcome 0 or 1 corresponding to thecollapse of the main register to one of the two states ^ (appropriatelynormalised). Figure 2 provides an example of a quantum circuit for performing an edge projec^on for 5an edge ^ = {^^, ^^} from the complement of a graph on 4 ver^ces. The projec^on includesplacing a Toffoli gate which is controlled using the qubits corresponding to ver^ces ^^and ^^and targeted on the auxiliary qubit. In the example of Figure 2, the auxiliary qubit is represented by |0> and the Toffoli gate comprises the inputs denoted ^^and ^^that control the NOT gate. The auxiliary qubit is then measured as indicated schema^cally by the box10 marked “M”, and the process is repeated as needed un^l the auxiliary measurement outcomeis 0. This edge projec^on of Figure 2 may be used to implement the simplex projec^on 150 of Figure 1. To generate the complete projec^on, ^^, for some Vietoris-Rips complex withunderlying graph ^, an edge projec^on may be performed for each edge ^ ∈ ^(^^ ). The15 method described in Akhalwaya et al. [2] performs this series of projec^ons while saving depth by construc^ng the complete projec^on circuit using the method shown in Figure 4. Here the edges in ^^are separated into sets of disjoint edges. For each such set, the corresponding edge projec^ons are performed in parallel. This construc^on uses as many Toffoli gates as there are edges in ^^.20 Figure 3 shows a schema^c example of the projec^on ^^ using edges (such as edges130 from Figure 1) using an approach proposed in Akhalwaya et al. [2]. The solid [black] linesrepresent afirst subset of edges from set 130, while the dashed [red] lines represent a second (disjoint) subset of edges. Again, the block for edges 130 represents classical processing, while the other two blocks in Figure 3 for quantum processing are denoted by circles in their 25 corners. The resul^ng projec^on is defined as a product of several par^al projec^ons, each of which removes edges belonging to a matching in ^^; this corresponds to complement ^^ 120as shown in Figure 1. (A matching is a collec^on of edges in which each edge is pairwise disjoint from all the other edges in the matching).QTDA 2 - QUK226739-WO; MJJ / 65238 PCTOne prac^cal problem with the above approach for obtaining and using clique data is the exponen^al increase in the number of poten^al cliques in a network as the size of thenetwork grows. For example, for a network having ^ nodes, the number of poten^al cliquesof size ^ is i.e. ^(^^), and this scaling o^en leads to classical algorithms which are very5 ^me-consuming (in some cases to the point of being imprac^cable). For this reason, there is significant interest in developing the use of quantum algorithms for compu^ng proper^esrelated to the cliques of a graph. Summary 10 The inven^on is defined in the appended claims. An apparatus is disclosed herein for preparing a quantum state represen^ng hypercliques of a D-hypergraph ^ which is formed from nodes connected by hyperedges Ewhose size is between 2 and afixed value D. The hypercliques comprise sets of nodes whosesubsets of size j between 2 and D are connected by hyperedges of size j. The apparatus is 15 configured to: generate a minimal hypergraph complement ^^and to use a classical computer system to preprocess the hypergraph complement intofirst and second subhypergraphs. The first subgraph comprises a D-simplex k-tree of hyperedges E’ from the hypergraph complementand the second subgraph comprising remaining hyperedges E’’ from the hypergraph complement. The apparatus is further configured to use afirst quantum circuit to prepare a 20 first quantum state from the D-simplex k-tree of hyperedges; and to use a second quantum circuit to prepare, based on thefirst quantum state and the remaining hyperedges, a second quantum state which stores in superposi^on the hypercliques of the hypergraph.An apparatus is disclosed herein for preparing a quantum state represen^ng cliques of a graph G which is formed from nodes connected by edges. The cliques of the graph comprise25 nodes that are all pairwise connected by edges. The apparatus is configured to generate agraph complement ^^ of the graph G and to use a classical computer system to preprocessthe graph complement intofirst and second subgraphs. Thefirst subgraph comprises a k-treeof edges E’ from the graph complement and the second subgraph comprises remaining edgesE’’ from the graph complement. Afirst quantum circuit is used to prepare afirst quantumQTDA 2 - QUK226739-WO; MJJ / 65238 PCTstate from the k-tree of edges. A second quantum circuit is used to prepare, based on thefirst quantum state and the remaining edges, a second quantum state which stores the cliquesof the graph in superposi^on. Addi^onal usages of thefirst and / or second quantum circuits are disclosed, as well as extending the applica^on to higher dimensionality (hypergraphs). 5 An apparatus is disclosed herein for preparing a warm start quantum state represen^ng hypercliques of a D-hypergraph ^ which is formed from nodes connected by hyperedges Ewhose size is between 2 and afixed value D. The hypercliques comprise sets of nodes whosesubsets of size between 2 and D are connected by hyperedges of size D. The apparatus is configured to generate a minimal hypergraph complement ^^and to use a classical computer 10 system to process the hypergraph complement into a subgraph comprising a D-simplex k-tree of hyperedges E’ from the hypergraph complement. The apparatus is further configured to use a quantum circuit to prepare the quantum state from the D-simplex k-tree ofhyperedges. An apparatus is disclosed herein for preparing a warm start quantum state represen^ng15 cliques of a graph ^ which is formed from nodes connected by edges E, the cliques of thegraph comprising nodes that are all pairwise connected by edges. The apparatus is configured to generate a graph complement ^^ of the graph and to use a classical computer system topreprocess the graph complement into a subgraph comprising a k-tree of edges E’ from thegraph complement. The apparatus is further configured to use a quantum circuit to prepare20 the warm start quantum state from the k-tree of edges.An apparatus is disclosed for performing a projec^on of a quantum state onto a subspace spanned by quantum states represen^ng hypercliques which exclude a given set of hyperedges H comprising sets H2, … HD, where Hi contains hyperedges of size i. The apparatus is configured to use a classical compu^ng system tofind, for each 2=< i =< D, a set of 25 hypercliques Ci of size >= i in the hypergraph whose hyperedges are the downward closure of those in Hi. The apparatus is further configured to construct a quantum circuit onfirst and second registers. For each i and each hyperclique K in Ci, a projec^on is performed on qubits of thefirst register onto the subspace spanned by quantum states, with a count of the overlapwith K less than or equal to i being stored in the second register. These projec^ons in seriesQTDA 2 - QUK226739-WO; MJJ / 65238 PCTachieve the desired projec^on onto the subspace of hypercliques which exclude thehyperedges H. Also disclosed herein are methods corresponding to the above apparatus. It is further disclosed that the skilled person may u^lise features corresponding to dependent claims in 5 conjunc^on with each other and / or with the above apparatus and / or methods without limita^on to the combina^ons specifically formed in the claims.In the current NISQ era before universal error correc^on, computer-implemented methods which are well adapted to the capabili^es and requirements of the quantum compu^ng 10 hardware are cri^cally important to extract the greatest efficiency from the available quantum compu^ng hardware. Example computer-implemented methods described herein have the advantage of being adapted to take account of the specific constraints and capabili^es of par^cular quantum compu^ng architectures. An example of a computer system architecture forwhich the configurability of the auxiliary register can be exploited is an ion-trap based quantum 15 computer such as, for example, Quan^nuum’s H1 and H2 ion-trap based quantum computers. The architectures of these systems provide high connec^vity between qubits within an ion trap -the connec^vity is not limited to connec^ons between nearest neighbours, and can be up toall-to-all connec^vity. In prac^ce, all-to-all connec^vity is not essen^al and the advantages of poten^ally high connec^vity are limited by the transport ^me to shu^le qubits into gate zones 20 (a region of the quantum computer in which gates may be performed on qubits). Therefore, for an op^mal implementa^on, the auxiliary qubits may be kept physically close to gate zones. This reduces the length of the transport ^me, enabling a shorter run-^me and improving the efficiency of processing. Furthermore, the size of the auxiliary register can be op^mized for the number of gate zones available. This would allow for the maximum possible parallelisa^on for25 the fewest necessary resources in the auxiliary register. Brief Descrip^on of the Drawings Figure 1 provides an overview of a known method described by Akhalwaya et al. [2] forconstruc^ng a quantum state encoding a Vietoris-Rips complex.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTFigure 2 provides an example of a quantum circuit for performing an edge projec^on such as may be used in the method of Figure 1. Figure 3 illustrates an example of genera^ng a quantum circuit for projec^on as part of the known method described by Akhalwaya et al. [2] for construc^ng a quantum state encoding5 a Vietoris-Rips complex. Figure 4 depicts a graph having ver^ces and edges to help illustrate some aspects of theprocessing described herein. Figure 5 shows an example of a method described herein for construc^ng a quantumstate encoding a Vietoris-Rips complex state.10 Figure 6 depicts the construc^on and opera^on of a warm start circuit as describedherein. Figure 7 is an overview of performing a quantum projec^on as described herein. Figure 8 depicts an example of compiling a projec^on onto the ^-simplices of a Vietoris- Rips complex using a method as disclosed herein. 15 Figure 9 depicts an example of a generalised method for construc^ng a quantum state as disclosed herein. Figure 10 is a diagram showing an example of the construc^on of a generalised warm start circuit for a generalised (^-simplex) ^-tree as described herein. Figure 11 is a diagram showing an example of the construc^on of projec^ons for20 removing cliques. Figure 11A is a schema^c diagram showing an example of a quantum compu^ng systemfor preparing a quantum state represen^ng cliques of a graph (and for other opera^ons as described herein). Figure 12 depicts an example of a circuit which may be used for crea^ng superposi^on25 over states of Hamming weights 0 or 1 as disclosed herein. Figure 13 depicts an example whereby for a given matching (M, le^), a state may beconstructed (right) to create a uniform superposi^on over states represen^ng subsets of thever^ces of a graph G which do not contain any edges from M.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTFigure 14 is an example of a circuit which prepares a superposi^on over states withHamming weight 0 or 1 on k-qubits as disclosed herein. Figure 15 illustrates a given collec^on of disjoint cliques (le^) from which a state (right)can be constructed which is a uniform superposi^on over states represen^ng subsets of the 5 ver^ces of the graph G which do not contain any edges from any of the disjoint cliques according to an approach described herein.Figure 16 is a circuit diagram showing an induc^ve step of the warm start algorithmaccording to an approach described herein. Figure 17 provides an example of construc^ng a quantum circuit (right) for a specific10 tree (le^), given a set of edges ^′ which form the tree ^, to generate a uniform superposi^onover states represen^ng subsets of the ver^ces which do not contain any edges from ^. Figure 18 presents an example showing a 3-tree on the le^ with its underlying tree structure on the right. Figure 19 depicts an example of a quantum circuit for loading a weighted superposi^on15 over ^-qubit states. Figure 20 shows an example of a quantum circuit for managing an induc^ve step for construc^ng a uniform superposi^on over a subsets of ver^ces.Figure 21 depicts a whole circuit (induc^ve) construc^on of ^^(^, ^) for a ^-tree ^with tree decomposi^on ^. 20 The drawings listed above are provided by way of example rather than by way of limita^on. Descrip^on a) Overview25 A network comprising (i) nodes and (ii) edges providing links or connec^ons betweenthe nodes provides a common form of rela^onal structure which is used to represent many forms of data, e.g. online social interac^ons, co-occurence of biomarkers, geometric proximity,and so on. The cliques of a network represent collec^ons of nodes which are, pairwise, all related to one another by direct edges. Cliques are o^en important features for modellingQTDA 2 - QUK226739-WO; MJJ / 65238 PCTdata and the interac^ons between cliques may be used to capture geometric and topological informa^on about the underlying network. One prac^cal problem with the use of clique informa^on is the rapid growth in thenumber of poten^al cliques as the network increases in size. For a network on n nodes, the5 number of poten^al cliques of size k is ^(^^). The approach disclosed herein supports theuse of a wide range of networks on n nodes, including how to prepare and project ontoquantum states on n qubits which store the cliques of the network in superposi^on. Measuring and manipula^ng these quantum states with other quantum circuits enables network analyses which extract topological and geometric informa^on about the cliques from10 the network. As described herein, a method may involve two classes of circuits: one for preparing quantum states based on a given network and one for projec^ng states onto the cliques of a given network. As described herein, classical pre-processing steps have been developed to inform circuit design and to help achieve greater efficiency. The two classes of circuits may be15 used together or separately. b) Terminology This sec^on summarizes some terminology used to describe the methods disclosedherein. (Some of this terminology is shared with the above descrip^on of the background and20 exis^ng work). Simplicial complexes: A simplicial complex is a data structure for describing rela^ons amongthe elements of a set ^ = {^^, ^^, … , ^^} of ^ nodes. More par^cularly, a simplicialcomplex is defined by a set Γ of simplices which are subsets ^ = {^^^ , ^^^ , … ^^^} of ^, suchthat Γ is closed under the subset rela^on (i.e. if ^ ∈ Γ and ^ ⊂ ^ then ^ ∈ Γ). As outlined25 above, this structure (simplicial complex) is important for describing many prac^cal rela^onsbetween data such as physical proximity and co-occurrence in interac^ons. This simplicialcomplex also represents a step for compu^ng topological invariants of data, such as Be^numbers (see, for example, h^ps: / / en.wikipedia.org / wiki / Be^_number).QTDA 2 - QUK226739-WO; MJJ / 65238 PCTWe say that a simplex ^ ∈ ^ has dimension ^ if |^| = ^ + 1; we call such a simplex a^-simplex. A simplicial complex ^ has dimension ^ if the largest dimension of any simplexof ^ is ^.A quantum computer may be configured to represent the simplices of an ^ node5 simplicial complex using computa^onal basis states on ^ qubits. The mapping fromsimplices ^ ⊂ … , ^^} to ^-qubit states is defined as ^ ↦ |^^ where |^^ is thecomputa^onal basis state |^^ … ^^^ such that, for each ^ between 1 and ^, ^^ = 1 if andonly if ^^ ∈ ^, and ^^ = 0 otherwise. Such a mapping for encoding simplices into qubitstates has been disclosed, for example, in Lloyd, Ganarone, Zanardi’s [9] algorithm for10 approxima^ng high dimensional Be^ numbers. Other encodings of simplices are available, see McArdle et al.

[0010] for an example. The method presented in this applica^on could be adapted to these other encodings if so desired, but the present applica^on will focus on theencoding described above (and associated examples).Given a simplicial complex Γ and an encoding of simplices into qubit states, an15 important quantum state for analysing Γ on a quantum computer is the state |Γ^ which is aquantum superposi^on over all states |^^ for which ^ ∈ Γ. That is, The present applica^on discloses various ways of preparing this state which improve on knownmethods.20 Graphs: A graph is a structure related to a simplicial complex and is denoted by ^. Forpresent purposes, a simplicial complex, also referred to as a downward-closed hypergraph, canbe considered as having structures that connect more than two nodes (ver^ces), rather than edges connected between pairs of ver^ces as for a conven^onal graph. Such a graph is defined as a pair of sets (^(^), ^(^)), where ^(^) is an arbitrary set25 whose elements are referred to as ver^ces and ^(^) is a collec^on of subsets of ^(^) of size2 whose elements are referred to as edges. This data structure is used in many applica^onsto represent networks. If we consider the edges of a graph as 1-simplices ^ ⊂ ^(^), a graphcan be seen as a 1-dimensional simplicial complex over the ver^ces of ^.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTFor a graph ^, the complement graph (also referred to herein as the graphcomplement) ^^ is a graph with the same set of ver^ces ^(^) and with edges ^(^)^. Aclique in a graph ^ is a subset of ver^ces such that for each pair ^ ^^ ∈ ^, thereis an edge {^^, ^^} ∈ ^(^). A matching in a graph ^ = is a subset of edges5 ℳ of ^ such that no two edges in ℳ share a common vertex. A perfect matching of ^ isa matching where every vertex of the graph is incident to some edge of ℳ. The Vietoris-Rips complex of a graph: For an arbitrary graph ^, a simplicial complex may bedefined on ^(^) called the Vietoris-Rips complex of ^, given byΓ^ = {^ ⊂ ^(^): ^ ^^ ^ ^^^^^^ ^^ ^}.10 We denote the Vietoris-Rips complex of a graph by Γ^throughout. A focus of the present applica^on is to provide a method for construc^ng the quantum state The approach described herein may be extended to higher dimensionality, for example, a hyperedge, a hyperclique, and so on. The terminology (hyper)edge is used herein to denote an en^ty which may be an edge or a hyperedge. Similar terminology is used with other 15 elements, such as a hyper(clique), which denotes an en^ty which may be a clique or a hyperclique. *** As an example to illustrate the above defini^ons, the diagram in Figure 4 represents agraph whose ver^ces and edges are given by the following sets:20 ^(^) = {1,2,3,4,5}; ^(^) = {{1,2}, {2,3}, {2,4}, {3,4}}.This graph yields a corresponding Vietoris-Rips complex given byΓ^ = {⌀, {1}, {2}, {3}, {4}, {5}, {1,2}, {2,3}, {2,4}, {3,4}, {2,3,4}},This complex defines the cliques of the graph from Figure 4, star^ng (by defini^on) with an empty set (no nodes / ver^ces), plus entries for all the single nodes, plus entries for each of25 the pairs of nodes (such as {2, 3} and {3, 4} defining edges), plus an entry for the clique definedby nodes {2, 3 and 4}. The quantum state associated with Γ^ is thefive qubit state (corresponding to thenumber of ver^ces):QTDA 2 - QUK226739-WO; MJJ / 65238 PCT |11000^ + |01100^ + |01010^ + |00110^ + |01110^)The network in Figure 4 can therefore be represented both as a graph and also as a corresponding Vietoris-Rips complex. 5 *** The above terminology can be generalised from graphs to higher dimensional rela^onalstructures, such as hypergraphs, hence the use in the applica^on of language such as “D- hypergraphs”. A hypergraph is a set of ver^ces V connected by a set of hyperedges H which are subsets of the set V. A hypergraph is called a D-hypergraph if its largest hyperedge has10 size D. This comes with subsequent defini^ons such as hypercliques, hypergraph complement and so on. This broader (generalised) language will be understood as encompassing the generalised structures defined below in the sec^on “Generalising to comple^on complexes”, in which alternate terminology is used to emphasise the generalisa^on of Vietoris-Rips complexes (defined for graphs) and comple^on complexes (defined for ‘k-skeletons, which are a higher 15 dimensional rela^onal structure). In the broadest generality, a k-skeleton, as described below, is a type of hypergraph. Further informa^on on terminology used herein is set out in Table 1 below: QTDA 2 - QUK226739-WO; MJJ / 65238 PCT Table 1 – Selected terminology (including higher dimensions)5 For a given row of Table 1, the terminology in thefirst column and the terminology in the second column are used interchangeably with one another (unless the differences in the third column are specifically of relevance). c) Quantum States10 The present applica^on describes, inter alia, a method for projec^ng a network onto aquantum state, whereby the quantum state stores the cliques of the network in superposi^on.Measuring and manipula^ng the quantum state with other quantum circuits allows a networkQTDA 2 - QUK226739-WO; MJJ / 65238 PCTanalysis to extract topological and geometric informa^on about the cliques from the graph. The method is suitable for use in conjunc^on with a wide range of networks on ^ nodes.An important aspect of this approach is the efficient prepara^on of the quantum state.Accordingly, a methodology is described herein for preparing a quantum state which is a 5 uniform superposi^on over the cliques of an arbitrary graph. In this approach, the desired state prepara^on circuit is wri^en as a product of two operators: (i) afirst determinis^c stateprepara^on circuit that prepares a quantum state that is close to the desired state – this isreferred to herein as a warm start state prepara^on, and (ii): a second quantum circuit implemen^ng a projec^on rou^ne which completes the prepara^on of the network cliques10 state. The quantum state produced by the warm start state requires a rela^vely low level offurther processing to reach the desired state (compared for example with the processing used by exis^ng techniques to reach such a desired state). Furthermore, a warm start state prepara^on is arranged to have a shorter depth and / or a higher probability of success than an exis^ng implementa^on, for example based on the configura^on of Figure 1 as described15 above. Various aspects of this approach provide advantages compared with exis^ng work. Some of these advantages may have poten^al applicability beyond the par^cular context described herein of coun^ng and / or analysing cliques in a network. For example, the ini^al warm start state prepara^on discussed above provides a stand-alone quantum circuit primi^ve20 that prepares a uniform superposi^on over computa^onal basis states which do not contain an edge of a ^-tree, which is a broad family of graphs that generalizes graph matchings, trees, and cliques. This state prepara^on rou^ne may also poten^ally serve as a useful subrou^ne within other state prepara^on methods. In addi^on, a projec^on rou^ne is described herein which provides a general method for preparing a uniform superposi^on over computa^onal 25 basis states which are not independent sets within the given graph. Furthermore, the present applica^on also discloses the use of several classical op^miza^on rou^nes to determine the op^mal use of these two subrou^nes to minimize the circuit depth of resul^ng circuits.A state prepara^on rou^ne as described herein may be compared, for example, to a projec^on onto the cliques of a graph according to Akhalwaya et al. in Ref. [2]. The approachQTDA 2 - QUK226739-WO; MJJ / 65238 PCTof Akhalwaya et al. [2] includes construc^ng a circuit of Toffoli gates targeted on an auxiliary register and measuring the auxiliary register and repea^ng the circuit if necessary un^l a par^cular measurement outcome occurs. Such an approach involves a number of Toffoli gates and repe^^ons which scale in propor^on to the number of edges present in the 5complementary graph, and therefore is only efficient for graphs that are sufficiently dense (interms of edges). By contrast, a method such as disclosed herein significantly reduces the number of Toffoli gates and repe^^ons required, thereby helping to reduce the overall ^meand complexity of implemen^ng the desired projec^on, and also providing significant savings incircuit depth in fault-tolerant se^ngs (given that the Toffoli gate is costly to use in a fault-10 tolerant implementa^on). In more general terms, a method such as disclosed herein may beimplemented without using a mul^-controlled NOT gate for each hyperedge, where mul^- controlled NOTs are a generalisa^on of Toffolis to higher dimensions which include hyperedges as a generalisa^on of edges. 15 d) State Construc^on Aprobabilis^c method is disclosed herein for construc^ng the state for theVietoris-Rips complex Γ with underlying graph ^. This method uses classical pre-processing530 and both state prepara^on and projec^on quantum circuits as described herein to producethe state The method generally uses fewer quantum resources than exis^ng work while20 providing a higher success probability. Figure 5 is aflowchart depic^ng an example of a method described herein for producinga Vietoris-Rips complex state (thisflowchart can be contrasted with Figure 1, which showsa known method for producing a Vietoris-Rips complex state As for Figure 1, in Figure 5the upper and middle [red] boxes or other shapes are based on classical compu^ng, while the25 two lower [blue] boxes with circles in their corners are based on quantum compu^ng.The opening por^on of the method shown in Figure 5 is the same as shown in Figure 1,in which an input network (graph) ^ 110 is provided, and the complement graph ^^ 120 isthen determined. In this complement graph, ^^ 120, pairs of nodes that are connected byan edge in ^ are not so connected in ^^ 120, conversely, pairs of nodes that are notQTDA 2 - QUK226739-WO; MJJ / 65238 PCTconnected by an edge in ^ are connected by an edge in ^^. For example, if we assume agraph ^ such as shown in Figure 4, then ^^ has the following edges: 1-3, 1-4, 1-5, 2-5, 3-5and 4-5. A^er the crea^on of ^^, the processing of Figure 5 diverges from the processing of 5 Figure 1. In par^cular, in Figure 5 classical preprocessing 530 is typically u^lised to split up the complement graph ^^120 into two edge-disjoint subgraphs, i.e. each edge from complement graph ^^ 120 is in one but not both of the subgraphs (although in some cases there may be anoverlap in edge membership between the two subgraphs, as discussed below). One of the subgraphs 535 in Figure 5 has the structure of a ^-tree and is used in a10 circuit for preparing a warm start state 540 as defined herein. The other subgraph 545comprises (at least) the remaining edges of ^^ and is used in the subsequent quantumprojec^on circuit 550 to produce the desired output state from the created “warm start”state 540 (as described in more detail below).15 1 – Classical preprocessingIn the example method of Figure 5, the classical preprocessing 530 sits at the interface between the classical compu^ng system and the quantum compu^ng system. The input to the classical preprocessing is the complement 120 of the graph data underlying the desired Vietoris-Rips state. The classical preprocessing 530 is configured to divide up the edges of the20 complement graph 120 into two (poten^ally overlapping) parts ^′ and ^′′ which aresubsequently used to construct two quantum circuits, namely the warm start circuit 540 and the quantum projec^on circuit 550. The classical preprocessing 530 is configured to split theedges into two parts (^′ and ^′′) in a way that helps to minimise the expected length of thecircuit for preparing ^Γ^^.25 The warm start circuit 540 , based on the set of edges ^′, is configured to create(determinis^cally) a quantum state |Γ′ ^ which is the quantum state corresponding to aVietoris-Rips complex defined on the graph ^′ which has the same ver^ces as ^ but hasedges defined by the rela^on = ^′. Thus ^′ is exactly the set of edges of thecomplement graph of ^′.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTIn order for the construc^on of the warm start circuit to be performed efficiently (with small gate count and depth), the edges ^′ are selected so that the graph has astructure amenable to a short quantum state prepara^on. The details of such quantum circuitconstruc^on are given below. Depending on which circuit construc^on is being used, the 5 classical pre-processing may try to iden^fy related structures in the complement graph ^^, e.g. cliques, ^-trees etc. Instances of such related structures can be generated using standard graph algorithms packages, e.g. NetworkX (see, for example, h^ps: / / networkx.org / ) , or may beproduced by other algorithms appropriate to the data domain (or provided directly by users). The apparatus of Figure 5 further includes a projec^on circuit 550 which is designed to10 prepare the desired state |Γ ^ from the warm start state |Γ′ ^ . This is achieved byperforming a series of entangling opera^ons and measurements involving a second register. The projec^on circuit of Figure 5 is similar in appearance to the projec^on circuit described inFigure 1 but has two significant differences. Firstly, the projec^on of Figure 5 does not useToffoli gates for every edge in ^′′, instead it groups these edges together into cliques and 15 performs simpler subprojec^ons (using shorter depth circuits with fewer auxiliary qubits) on these cliques. Secondly, the projec^on circuit 550 of Figure 5 is informed by the knowledgethat its input will be the state |Γ′ ^ from the warm start circuit 540 rather than a generaluniform state. As shown in more detail below, these techniques (differences) generally result in a large 20 saving in resources. Furthermore, the success probability in the projec^on described in the prior art isfixed by the overlap between the uniform state on ^ qubits |^ ^ and the desiredstate |Γ ^ , specifically the inner product ||^Ψ|Γ^||^. In contrast, the probability of success inthe method described herein is || which can be controlled by the classical processing circuit 530. As this projec^on is to be performed as a series of rounds with some probability 25 of failure in each round, the depth of circuit executed will vary depending on if and when the projec^on fails. The depth of the warm start circuit 540 is denoted as ^^^^ and ^^^^,^^^ is used todenote the expected depth of circuit involved before thefirst successful projec^on to the state |Γ ^. The sets ^′ and ^′′ are chosen so as to reduce the total length of a quantum circuitQTDA 2 - QUK226739-WO; MJJ / 65238 PCTwhich is expected to be run before the state |Γ ^ is successfully obtained. This expected costcan be calculated for a given ^′ and ^′′ as The preprocessing step then comprises genera^ng viable decomposi^ons of the set5 ^(^^) into sets ^′ and ^′′ using one or more of the available classical methods (subject tothe available classical compu^ng resources) and then choosing a decomposi^on whichminimises the cost ^(^′, ^′′). This process is dis^nct from the process of quantum circuitcompila^on which typically does not have direct access to the underlying data being analysed (in this case ^) and does not have control over projec^ons as proposed herein. The circuits10 resul^ng from the above process may be sent to a compiler and this may further reduce theresource requirements. We now describe the components of this warm start circuit 540 in more detail.2 – Warm Start Circuit15 The warm start circuit 540 is used to produce a uniform quantum state |Γ′ ^ which canbe projected onto the desired quantum state |Γ ^ with a quantum circuit which has a shorterdepth and / or a higher probability of success than the quantum circuit used to project to |Γ ^from a uniform superposi^on |Ψ ^ over all ^-qubit states (such as depicted in Figure 1). Asnoted in exis^ng work on uniform state prepara^on

[0011] , generally it is not expected to be20 feasible to prepare an arbitrary uniform state |Γ′ ^ using short depth circuits. However, asdisclosed herein, certain quantum states may be prepared with efficient determinis^c quantumcircuits and these quantum states are used herein as the basis for the warm start methodillustrated in Figure 5. The present approach recognises that for a certain class of graphs, any^′ on ^-ver^ces in this class gives rise to an efficient state prepara^on method for the uniform25 quantum state on ^-qubits which excludes an edge in the graph ^′. A^erfixing some targetgraph class for which efficient prepara^ons are known, the following method may be adopted:1. Given the set of edges ^,find (using classical op^misa^on) the largest set ^′ whichforms a graph from the target class.QTDA 2 - QUK226739-WO; MJJ / 65238 PCT2. Construct the efficient state prepara^on quantum circuit for the state |Γ′ ^ which isuniform over quantum states which exclude edges from ^′. 3. Perform a projec^on on |Γ′ ^ which removes all edges from ^\^′, thus forming thedesired state |Γ ^.5 Note that in the above method, the determina^on of the largest set may be subject to one or more criteria. For example, the largest set E’ may be determined as the largest set located from the set of edges within a given search procedure (such as within a maximum number of itera^ons, a given processing ^me, and so on). The largest set may also be determined, for example, as thefirst set which exceeds a given threshold (even if it is possible10 that the set E may contain larger sets for E’). Accordingly, it will be understood that in some implementa^ons, the selec^on of the largest set may incorporate one or more prac^cal criteria (as opposed to adop^ng a purely mathema^cal defini^on for the largest set E’. The Appendix below gives examples of different target graph classes that provide efficient state prepara^on circuits for |Γ′ ^ and efficient state prepara^on circuits are15 described. These graph classes include: • Matchings • Sets of disjoint cliques • Trees • “^-trees” (which are trees of cliques)20 Running the warm start algorithm as described herein u^lises a method of reliablyfinding the structures in a given graph ^. For both matchings and trees, efficient algorithmsforfinding maximal examples in a graph are known. For cliques, the naïve problemcomplexity scales exponen^ally with the size of cliques targeted, however there are performant open source algorithms for these such as included in NetworkX [6]. A significant aspect is that25 the warm start described herein is able to work with (and deliver savings for) even sub-op^malclique iden^fica^on that is reachable in polynomial ^me. Such algorithms can reliablyfind large cliques in graphs having large numbers of ver^ces. ^-trees, which are constructed fromoverlapping cliques, may be found by adap^ng available cliquefinding algorithms.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTFigure 6 is a diagram illustra^ng an example of the construc^on of a warm start circuit540 as described herein. Consistent with other diagrams herein, the [blue] box to the rightwith circles in its corners represents a quantum compu^ng resource, while the [red] box to thele^ represents a classical (determinis^c) compu^ng resource. The construc^on u^lises k-tree 5 535 shown in the le^-hand (classical) sec^on of Figure 6. The ver^ces of the k-tree formed from G’ are arranged into a collec^on of overlapping subsets which are represented by the nodes of a tree. Each of these nodes represents (is) a clique in the graph G’.In the right-hand sec^on of Figure 6, the k-tree structure is converted into a warm startcircuit 540 which prepares a quantum state |Γ’^ which is a uniform superposi^on over all10 quantum states represen^ng subsets of the ver^ces which do not contain any edges from G’. The structure of the quantum circuit follows that of the k-tree 535, with each node genera^ng aquantum subcircuit which excludes all edges in the clique represented by the node. The tree structure allows these subcircuits to be connected together to create the warm start circuit 540 for |Γ’^.15 The projec^ons in the next step of the method will now be described.3 - Simplex Projec^onA^er the warm start circuit 540 has created the state |Γ′ ^ as described in the previoussec^on, a projec^on quantum circuit 550 is run to create the desired simplex state |Γ ^. This20 involves taking the set of remaining edges ^′′ and crea^ng a circuit which separates between(i) states which contain any of these edges, and (ii) states which contain none of these edges. Exis^ng solu^ons have performed such a projec^on by using separate Toffoli gates for each of these edges as discussed above. The approach described herein has various benefits over such exis^ng solu^ons. For25 example, it is shown herein that iden^fying cliques in ^′′ allows us to use Hamming weightprojec^ons instead of Toffoli gates to perform the desired projec^ons. These Hamming-weight based projec^ons in general have significantly shorter circuit depth than those produced using Toffoli gates. In addi^on, there is a further reduc^on of the resource requirements forthe projec^on described herein if we know in advance a set ^′ of edges which have alreadyQTDA 2 - QUK226739-WO; MJJ / 65238 PCTbeen removed from the state. This results in savings par^cularly when this projec^on is performed a^er the warm start circuit 540. The degree of savings varies according to the(hyper)graph which is being processed. An overview of this method is depicted in Figure 7. As for Figure 6, the [blue] boxes to5 the right with circles in their corners represent quantum compu^ng resources, while the line [red] box to the le^ represents a classical (determinis^c) compu^ng resource.Figure 7 depicts a set E of edges 735 and selec^ng a set of disjoint cliques covering as many edges as possible for removal to create the next (par^al) projec^on 750 en route to thefull projec^on 790. Figure 7 illustrates how such an edge projec^on as described herein is 10 able to improve upon exis^ng work, such as the projec^on shown in Figure 1. For the present applica^on, rather than use a product of par^al projec^ons which removes one edge of the complement graph at a ^me, the method described herein (and shown in Figure 7) removes aclique from the complement graph in each round of processing.Considering the present approach in more detail, there is a shi^ in focus from edges (as15 per Figure 1) to cliques (as per Figures 5, 6 and 7). The quantum circuit construc^on methoddescribed herein begins byfinding the largest (or at least rela^vely large) cliques in ^′′ andarranging them into rounds of disjoint cliques. Then for each such disjoint clique it is known that the states on the qubits represen^ng ver^ces of this clique can be readily separated. A subset of these ver^ces contain an edge from ^′′ if and only if the set contains more than one20 element ( this is because the ver^ces form a clique). Accordingly, a computa^onal basis state|^^ … ^^^ on the corresponding qubits should be accepted if and only if the Hamming weight∑ ^^ is less than 2.Hamming weight projec^ons are a common tool in the literature. This exact projec^on can be done in a bitwise manner with ^(^) auxiliary qubits in two-qubit gate depth25 ^((log^)^) and ^(log^) measurements [1]. This is an improvement on the Toffoli methodas described above which uses ^(^) auxiliary qubits to perform the projec^on in two-qubitgate depth measurements. U^lising the approach of [1] may thereforerepresent an exponen^al saving of resources in terms of gate depth and measurements foreach clique in ^′′.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTA further aspect of the approach described herein is the adaptability of the quantum projec^on circuit 550 depending on knowledge of the input. Such knowledge of the input state is common in many applica^ons of the quantum projec^on circuit. For example, if we prepare the state |Γ ^ according to the approach described herein, it is known that the input5 state to the projec^on circuit is |Γ′ ^, which is not a general superposi^on of computa^onalbasis states but rather is restricted to a subspace ℋ′ of the Hilbert space on ^ qubits.This subspace is defined by the set ^′ of edges in ^^ which have been excluded from |Γ′ ^by running the warm start 540.This knowledge can now be used to adapt the bitwise Hamming weight projec^ons in10 the new ^^ described above. According to the bitwise Hamming weight projec^on [1] ascited above, performing this projec^on on ^ qubits involves a series of projec^ons for each ofthe ⌈log^^ + 1⌉ bits used to store the Hamming weight on the ^ qubits. However, as notedin [1], if the input state is in a restricted Hamming weight subspace where the maximum Hamming weight is ^′, then only ⌈log^′ + 1⌉ projec^ons may be u^lised, thereby reducing the15 circuit depth to ^(log^ × log^′) and the number of measurements to ^(log^′)measurements. When ^′ is small rela^ve to ^ this represents a substan^al saving. Accessing thissaving involves a slight modifica^on to the method described above, in that instead of searching for disjoint cliques in ^′′ in each round of projec^on, a search is made for a set of20 disjoint cliques in the whole ^^. Even though some of the edges of these cliques will already be accounted for by the warm start circuit 540 or by previous rounds of the projec^on, this isgenerally outweighed by the savings from adap^ng the bitwise projec^ons. This may be done as follows. Firstly, denote ^^ the set of edges in ^^ which havealready been excluded from the quantum state and let ^^^ , … be a collec^on of disjoint 25 cliques in ^^containing a large number of edges in ^(^^)\^^. Then for each clique ^^^of size ^^, compute ^′^, the minimum number of cliques in ^^required to cover the ver^ces of ^^^. As single ver^ces are always cliques (1-cliques), we have that ^′^ ≤ ^^. This leads to thepossibility of reducing the number of rounds in each bitwise Hamming weight projec^on.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTFurther efficiencies can be gained by persis^ng some of the Hamming weight projec^on circuitry between rounds of this projec^on. This works as follows: in Round i of the projec^on process described in Figure 7, we perform a Hamming weight projec^on ^^^for each clique ^^ ^in Round i and we compute some bits of the Hamming weight which are then uncomputed in5 the event of a successful computa^on, see [1]. This process of uncomputa^on typically requires⌈log^^ + 1⌉ controlled z-gates for each vertex in the clique ^^^. We can reduce this burden by matching the cliques {^^^,…,^^^^ } with those in the next round {^^^^^,… ^^^^^^^^}. Such a matching {(^^, ^^), … , (^^ , ^^)}, where the ^^ values are dis^nct whole numbers between 1 and andthe ^^ values are dis^nct whole numbers between 1 and ^^, describes how to pass on the10 Hamming weight totals computed in Round i for the start of Round i + 1. For example, if the pair(j, ^ ) is in t ^^ he set of matching cliques and if the projec^on for ^^ succeeds, then instead ofuncompu^ng all of the ver^ces of ^^^, we only uncompute the ver^ces corresponding to the ver^ces not in ^^^^^^ . Similarly, when we start the ^^^ projec^on in Round j+1, correspondingto , we only need to add ver^ces from^ which do not appear in ^^. This saves a15 total of controlled z-gates. We can choose such a matchingapproach which aims to maximise the sum of the sizes of the intersec^ons ^^^ ∩ ^^^^^^. *** e) Extensions of the above Methodology This sec^on describes two extensions to the method described above. Thefirst20 extension is specific to the case where we care about ^-simplices of the complex. The second extension relates to a more general set of complexes than the Vietoris-Rip complex which can be dealt with using a modified version of the above method. Restric^ng to k-simplices 25 There are several important algorithms which include a projec^on onto the ^-simplices of a Vietoris-Rips complex. For example, Akhalwaya et al. [2] use a projec^on onto the ^-simplices as a subrou^ne for their Be^ number es^ma^on quantum algorithm. In general, aprojec^on may compile onto the ^-simplices by expressing the projec^on as a product of twoQTDA 2 - QUK226739-WO; MJJ / 65238 PCTprojec^ons ^^and ^^, namely the projec^on onto the full Vietoris-Rips complex and the projec^on onto the Hamming weight ^ states, respec^vely. Each of the two projec^ons isthen compiled separately from the other projec^on. As disclosed herein, this compila^onincludes an addi^onal classical pre-processing step on the Vietoris-Rips complex to simplify the 5 compila^on of the composite projec^on ^^^^. This approach helps to ini^ally eliminate some trivial instances that might occur before proceeding to more complicated compila^on techniques. Is ^^^^ equal to 0? If the simplicial complex Γ does not contain any ^-simplices,this indicates that ^^^^is equal to 0. Any quantum algorithm which uses ^^^^as a 10 subcircuit in this case will then be trivial, and therefore it is helpful to perform some efficient classical algorithms on the underlying graph to determine if this is the case before proceeding to do the more costly compila^ons of ^^and ^^. In par^cular, there are a few simple proper^es of the underlying graph which can be efficiently computed and which imply thatthere are no ^-simplices. For example, if the graph contains a ^-clique then it contains ^15 ver^ces with degree at least ^ − 1. Similarly, if ^ ≥ 3 and the graph contains a ^-cliquethen the graph cannot be bipar^te. The approach described herein therefore u^lises a package of standard graph algorithms such as NetworkX to determine if the graph is bipar^te, or if there are fewer than ^ ver^ces with degree at least ^ − 1. If either of these cases isfound, then it is determined during this pre-processing step that ^^^^ is equal to 0 and this20 finding avoids performing a compila^on of ^^or ^^. Reducing the number of effec^ve qubits. There are many known quantum algorithms which compute certain proper^es of a simplicial complex that depend only on the connected components. For example, the 25 quantum algorithm presented in Akhalwaya et al. [2] gives an es^mate for the Be^ number of a simplicial complex. The Be^ number of a simplicial complex is equal to the sum of the Be^ numbers of its connected components, and therefore the quantum algorithm can alterna^vely be run over the simplicial complex of each connected component and then the es^mates for each component are summed together. This la^er approach may be beneficial as it reducesQTDA 2 - QUK226739-WO; MJJ / 65238 PCTthe number of qubits used to run a suitable algorithm, par^cularly over simplicial complexeswith many connected components. For applica^ons of this form, a graph algorithms package such as NetworkX may be used as a pre-processing step to compute the connected components of the underlying graph. 5Following this step, a compila^on may be performed of the simplicial complex statescorresponding to each connected component as described herein. Figure 8 illustrates an example of a complete pipeline for compiling the projec^on ontothe ^-simplices of a Vietoris-Rips complex as described in Sec^on 5.1. Note that the central components of Figure 8 (in box 800) correspond to matching components in Figure 5, namely10 providing a graph complement 120, classical processing 530, spli^ng into the k-tree 535 andthe remaining edges 545, performing a warm start 540 using the k-tree, and performing a quantum projec^on 550 based on the warm start 540 and the remaining edges 545. The procedure of Figure 8 varies from Figure 5, at the input and output stages. At theclassical input stage (boxes to the le^, full outline in red) a graph ^ and parameter k are15 provided (opera^on 802) and addi^onal classical pre-processing is performed at opera^on 804to simplify ^^before proceeding to the rou^ne described in Figure 5. Likewise, the processing of Figure 8 has an addi^onal projec^on (opera^on 890) in rela^on to the (quantum)output from system 800 to a^ain |^^ >.20 Generalising to comple^on complexes A^-skeleton Δ is a simplicial complex on ^ ver^ces (0-simplices) in which the highestorder simplex has order ^. It is given as a list of sets Δ^, Δ^, … Δ^ where Δ^ is the set of all^-simplices of Δ. In this sense, an undirected graph ^ can be thought of as a 1-skeleton,where ^^is the set of ver^ces and is the set of edges. Given a ^-skeleton Δ, the25 comple^on complex of Δ is the complex Δ^ whose ^-simplices for ^ ≤ ^ are given by Δ^, i.e.the ^-simplices of the ^-skeleton and for ^ > ^ are all sets ^ = … ^^} for which∀^′ ⊂ ^ of size |^′| = ^ and we have ^′ ∈ Δ^. In this sense, the comple^on complex ofany graph ^ is the Vietoris-Rips (VR) complex of ^. These complexes are important becauseQTDA 2 - QUK226739-WO; MJJ / 65238 PCTthey can be used to approximate the Čech complex [7] which is important in data science applica^ons

[0012] . As described above, the simplices of a VR complex with underlying graph ^ are exactlythose subsets of the ver^ces of ^ which do not contain any edge from the complement graph5 ^^ . This observa^on supports the method shown in Figure 5 for preparing the VR complexstate, which divides up edges in ^^ to be ‘excluded’ from the quantum state by the warm startalgorithm or the quantum projec^on method. In the generalised case with a ^-skeleton Δ, it can be seen that the simplices of Δ^ areexactly those subsets of Δ^which do not contain any of the sets in Δ^^ , Δ^^ , … , Δ^^ where Δ^^ is10 the set of all subsets of Δ^ of size ^ + 1 which are not in Δ^. One subtlety compared to theVR method is that there is some redundancy in the collec^on of simplices described by Δ^^ , Δ^^ , … , Δ ^^ . In par^cular, if there is some ^ ∈ Δ ^^ , then for any ^ > ^, every subset of Δ^of size ^ which contains ^ will be in Δ^^. As a result, the complement structure can be completely characterised by ignoring these elements which contain elements from earlier15 complements. We refer to the remaining elements as the minimal complement simplices anddenote them with the sets Δ^^ , Δ^^ , … ,^ Δ^; they can be computed efficiently by star^ng withΔ^^ = Δ^^ and then for every ^ > 1, removing any set from Δ^^which contains any set from Δ^^for ^ < ^.The method in the above Sec^on ‘State Construc^on’ depicts an example of20 construc^ng the state |Γ ^ for the VR complex of a graph by ‘removing’ any edges in ^^ via amixture of warm start state prepara^on and quantum projec^ons with the division of edges decided in pre-processing. Similarly, the generalised method in the sec^on ‘Generalising tocomple^on complexes’ constructs a state for the comple^on complex of a ^-skeleton Δ byfirst preprocessing the minimal complement simplices Δ^^, Δ^^, … , Δ^^ so that some are handled25 by a generalisa^on of warm start Δ^^ and others, Δ^^ for each 1 ≤ ^ ≤ ^, which are handledby a generalisa^on of the quantum projec^ons method. As described above, this spli^ng is handled by a classical preprocessing step. A similar equa^on for the expected circuit length given a choice of Δ^^ , Δ^^, … , Δ^^ can be determined and the choice may be op^mised bygenera^ng choices of the Δ^ sets which are viable with the available classical compu^ng powerQTDA 2 - QUK226739-WO; MJJ / 65238 PCTand choosing one with the shortest expected depth. Over several rounds of such projec^ons, these savings are likely to be significant and can help to exploit extra structure in the input data. Figure 9 depicts an example of such a generalised method for construc^ng the quantum state represen^ng the comple^on complex of a given ^-skeleton Δ. The process shown in5 Figure 9 broadly corresponds to that described in Figure 5, but the process of Figure 9 isgeneralised to deal with the more complicated input and target state. Consistent with other diagrams herein, the [blue] boxes in the lower por^on of the diagram with circles in theircorners represent a quantum compu^ng resource, while the [red] boxes in the upper por^on ofthe diagram represent a classical (determinis^c) compu^ng resource.10 In thefirst opera^on of Figure 9, the sets of minimal complement simplices Δ^^, … Δ^^are constructed. These sets are split up into two parts: thefirst having the structure of a generalised ^-tree (or other similar structure) and the second having the remaining minimalcomplement simplices. The desired state is then created in two steps. Firstly a determinis^c ‘generalised warm start’ is used to prepare a uniform state which excludes all minimal15 complement simplices assigned to thefirst group. This component is described in more detail in Figure 10 below. Secondly, a series of probabilis^c projec^ons are applied to remove thecomplement simplices in the second group. This component is described in more detail in Figure 11 below.20 1- Generalised Warm StartConsider preparing the state represen^ng the comple^on complex of a ^-skeleton Δwhose minimal complement simplices are given for each dimension by the sets ^ Δ^, Δ^^, … , Δ^^ .We want to show that, for certain substructures of this complement structure, a state can beprepared which excludes all simplices in the substructure (in the same way as shown below in25 the Appendix for substructures of graphs such as matchings, trees, cliques and ^-trees).To detail how warm start works for the generalisa^on of cliques and ^-trees, we define aclique in Δ^^ to be a set ^ of ver^ces of Δ (i.e. elements of Δ^) such that for each subset^ ⊂ ^ of size ^ + 1 we have ^ ∈ Δ^^ (this is a direct generalisa^on of the defini^on of a cliquein a graph). Suppose ^^, ^^, … ^^ are sets of cliques in Δ^^, …^ Δ^. We can generate cliquesQTDA 2 - QUK226739-WO; MJJ / 65238 PCTin Δ^^ using a generalised version of the greedy algorithm for graph cliques. Namely, given aclique C of size k in Δ^^, we can check whether some new vertex v can extend this to a clique ofsize k + 1 by checking if v ∪ S is a clique for each S ⊂ C of size i.Given any clique ^ ∈ ^^ , a quantum circuit can be constructed on the qubits for the5 ver^ces of ^; the quantum circuit prepares a uniform superposi^on over all states excludingthe ^-simplices in ^. This is the quantum circuit loading the Dicke states of Hamming weight less than or equal to ^, generalising the quantum circuit given in Figure 14 below (suitablequantum circuits are known in the literature, see for example [4]). For any disjoint union ofcliques from any Δ^^the warm start state can be prepared by running these quantum circuits in10 parallel. To generalise the algorithm for ^-trees from the Appendix, the defini^on of a ^-tree in Δ^^, … Δ^^ is generalised to be a tree structure in which each node ^ represents a clique insome Δ^^^ and each vertex in Δ only appears in the cliques for a connected subset of thenodes. Note that for the ver^ces in the clique at node ^, the computa^onal basis states15 which are ‘allowed’ at this node are exactly those with Hamming weight ≤ ^^. At each node aweighted superposi^on is prepared over these possible states. Arbitrary weighted superposi^ons can be prepared with quantum circuits of depth poly-logarithmic in the number of allowed basis vectors [3], which in this case is at most ^(^^), giving polylog ^ circuitswhen ^ isfixed. The weights at each node are determined by the recursive algorithm which20 computes the number of acceptable warmstart states on all ^-qubits that result from extending the given state on ^^. To assemble the generalised warmstart circuit for such a generalised ^-tree, the state prepara^on circuits at each node are connected with those at its children. As the weighted prepara^on circuit of the child depends on the number and posi^on of |1 ^ states in the qubits25 represen^ng the intersec^on between the cliques at parent and child nodes, a condi^oned prepara^on circuit is used in the child node. Figure 10 is a diagram showing an example of the construc^on of the generalised warmstart circuit for a generalised (^-simplex) ^-tree as described herein. The system shown inFigure 10 is an adapta^on of the system in Figure 6. In par^cular, Figure 10 has a d-simplexQTDA 2 - QUK226739-WO; MJJ / 65238 PCTtree 1035 corresponding to the k-tree 535 of Figure 6 and a warm start circuit 1040 corresponding to the warm start circuit 540 of Figure 6. The generalised (d-simplex) ^-tree 1035 of Figure 10 comprises nodes represen^ngcliques in Δ^^ for some dimension ^, where each ^ is denoted in Figure 10 by a corresponding5 colour. In par^cular, Figure 10 depicts four different dimensions: Δ1, [black], corresponding to nodes E and F; Δ2, [red], corresponding to nodes A and B; Δ3, [blue], corresponding to nodeD; and Δ4, [green], corresponding to node C. The same colours are u^lised in the depic^on of the warm state circuit 1040 with the addi^onal use of pa^erned backgrounds: slopes up from le^ to right [red]; slopes down from le^ to right [blue]; horizontal lines [green]; and a solid light10 grey [black]block. In the circuit of Figure 10, the blocks coming from cliques of dimension ^ nowrepresent weighted loaders over all states of Hamming weight ≤ ^. The controlledopera^ons between (i) blocks represen^ng parent nodes and (ii) blocks represen^ng their children, are used to modify the weights of the child loaders, condi^onal on the state of the15 intersec^on between child and parent cliques. 2– Generalised Simplex Projec^onsThe projec^ons act to project onto the subspace of the ^-qubit Hilbert space spanned20 by those computa^onal basis states represen^ng the simplices of the comple^on of some ^- skeleton Δ. This is a direct analogy to the projec^on onto the space spanned by VR simplices described in the above sec^on “Simplex Projec^on”. Naively, this projec^on can be performed by doing a projec^on for each minimal complement simplex ^ in Δ^^. These projec^ons would generalise the edge projec^on given25 in Figure 3 by performing a gate controlled on all the qubits represen^ng ver^ces of ^.Here we describe a similar improvement to that described in the above sec^on “Simplex Projec^on” (compared with the naive method). Suppose that ^ is a clique of size ^ in Δ^^, as defined above. To perform theprojec^on which removes all states containing any ^-simplex in ^, the naïve method u^lisesQTDA 2 - QUK226739-WO; MJJ / 65238 PCT^^^^^ gates and the same number of measurements. However, generalising the approach used in the above sec^on “Simplex Projec^on”, it can be seen that the computa^onal basis states on the qubits represen^ng the ver^ces of ^ which survive this sequence ofprojec^ons are exactly those which have Hamming weight ≤ ^. This projec^on can be5 performed using a bitwise Hamming projec^on as described in [5] and u^lises just ^(log^^)gate depth and ^(log^) measurements, significantly outperforming the naive method on sucha clique. To assemble the full circuit for the projec^on which excludes all simplices in Δ^^, … , Δ^^,first a set of cliques are computed in each dimension ^^, … , ^^ such that each minimal10 complement simplex is in at least one clique. We canfind such cliques using classical techniques as described above, with bigger cliques (and thus bigger circuit savings) requiring more classical compu^ng power. These cliques are then arranged into rounds of cliques which are disjoint from one another. The projec^on circuits for the cliques in each round can thus be performed in parallel and the collec^ve effect of performing all of these rounds is to 15 perform the correct projec^on. This projec^on method can be further adapted when used a^er the generalised warm- start circuit described above. In par^cular, let ^ be a clique on ^-ver^ces for a Hammingweight ≤ ^ projec^on performed in one of the rounds described above. Let ^^ be the setof cliques which have been removed in previous rounds or in the warm start. Each of these20 cliques has a dimension ^ which is the Hamming weight of any state on the ver^cesrepresen^ng this clique. Suppose wefind a collec^on of cliques ^^, ^^, … ^^ which cover allthe ver^ces in ^ and let ^′ = ^^ + ⋯ ^^ be the sum of their dimension. (We can searchfor such collec^ons rela^vely efficiently by itera^ng through the list of cliques which overlap with ^). As noted in the above sec^on “Simplex Projec^on”, the Hamming weight ≤ ^25 projec^on can be adapted to use gate depth ^(log^′ × log^) with ^(log^′) measurements.Figure 11 is a diagram showing an example of the construc^on of projec^ons for removing cliques and can be regarded as a modifica^on or enhancement of Figure 7 using t-simplices 1135 and par^al projec^ons 1150 to create a full projec^on 1190. Consistent withQTDA 2 - QUK226739-WO; MJJ / 65238 PCTother diagrams herein, the [blue] boxes to the right with circles in their corners represent aquantum compu^ng resource, while the line [red] box to the le^ represents a classical(determinis^c) compu^ng resource. In par^cular, Figure 11 depicts the construc^on ofprojec^ons for removing cliques in Δ^^ for some dimension ^, by means of a sequence of5 par^al projec^ons. For each clique in Δ^^, a projector 1150 is placed for restric^ng toHamming weight ≤ ^ on the corresponding qubits. In each round of this loop, the largestpossible set of disjoint cliques across Δ^^ for different values of ^ are collected, as these canbe processed in parallel on the quantum circuit. In some implementa^ons, the system is configured to iden^fy trivial circuit10 implementa^ons of a projec^on onto the set of (hyper)cliques of a (hyper)graph when the projec^on is followed by a projec^on onto the (hyper)cliques of a given size s. In such an implementa^on, the iden^fica^on of trivial circuit implementa^ons may be based, for example, on iden^fying a par^cular class of hyper(cliques) having known mathema^cal proper^es to facilitate rapid implementa^on. 15 f) Performance ComparisonThe table (Table 2) below provides comparisons between resource requirements with and without warm start. The comparisons provide expected numbers of runs of quantum circuits with or without warm start (WS) for crea^ng a uniform superposi^on |G> over states 20 represen^ng the cliques of G where G is a complete graph (all-to-all connected) with a structure removed. The value of ϕ=(1+√5) / 2. 32QTDA 2 - QUK226739-WO; MJJ / 65238 PCT TABLE 2 g) Implementa^on Aspects Figure 11A is a schema^c diagram showing an example of a quantum compu^ng system5 such as for preparing a quantum state represen^ng cliques of a graph (and for other opera^ons as described herein). The compu^ng system of Figure 11A comprises two components, aclassical compu^ng system 210 and a quantum computer system 250. The classical compu^ng system 210 may comprise a known form of digital computer(s) including one or more processors for execu^ng program instruc^ons and memory for storing the program 10 instruc^ons and data. Note that in some cases, the quantum computer system 250 may be provided by an emula^on of a quantum compu^ng system running on (provided by) a classical computer system 210. Such emula^on may be used, for example, when developing aprogram for use on a quantum compu^ng system, to allow tes^ng of the program in a noise- free environment.15 The classical compu^ng system 210 is shown as including two facili^es, namely acompiler 220 and a control facility 225. These two facili^es are shown for convenience as located on a single classical compu^ng system 210, but they could be provided on two or more separate classical compu^ng systems if so desired. The compiler 220 is responsible for taking as input program (so^ware) instruc^ons and implemen^ng the instruc^ons on the quantum20 computer. The control facility 225 also may be used to provide a user with control over theopera^on of programs on the quantum compu^ng system 250. For example, the control facility 225 may allow a user to specify se^ngs for a program which are then applied duringexecu^on of the program on the quantum compu^ng system 250. The control facility 225may also be used to manage various interopera^ons between the classical compu^ng system 25 210 and the quantum compu^ng system 250, for example, transferring a compiled quantum circuit (program) to the quantum compu^ng system 250 for execu^on.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTThe quantum compu^ng system 250 includes at least one quantum circuit 260, which isconfigured to interact directly with the hardware of the quantum compu^ng system, for example to create and manipulate qubits 255. The quantum compu^ng system 250 further includes various gates 256 for performing opera^ons on the qubits 255. The quantum circuit 5 260 can be considered as somewhat analogous to a compiled program (low-level code) which has been adapted to run on the specific hardware implementa^on of the quantum computer, such as reflec^ng the number and connec^vity of the qubits 255 and gates 256 available on thequantum computer. The quantum compu^ng system 250 of Figure 11A further includesfirst and second10 registers 281, 282 for storing qubits between performing opera^ons with gates 256. Although thefirst and second registers 281, 282 are shown separately in Figure 9, they may be implemented (for example) using a single register having afirst por^on corresponding to the first register 281 and a second por^on corresponding to the second register 282. Moregenerally, it will be appreciated that the configura^on and architecture shown in Figure 9 is 15 provided by way of illustra^on and not by way of limita^on and hence the approach described herein may be implemented on many different types of quantum compu^ng systems or pla^orms. With reference to a par^cular configura^on such as shown in Figure 5, the opera^ons toreceive an input network 110, to generate the graph complement 120, to perform the classical 20 preprocessing 530, and to split into the k-tree 535 and the remaining opera^ons 545, are performed on the classical compu^ng system 210, such as by using the control facility 225, and more par^cularly by execu^ng one or more processors to implement the control facility. Thewarm start circuit 540 and quantum projec^ons 550 as shown in Figure 5 are generallyimplemented on the quantum compu^ng system 250 of Figure 11A (subject in some cases to25 instruc^ons from the control facility 225). During such opera^ons, the warm start circuit 540 and quantum projec^ons 550 may u^lise the resources available on the quantum compu^ngsystem 250, such as qubits 255, gates 256 andfirst and second registers 281, 282, to perform the desired opera^ons as described herein.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTAPPENDIX Some classes of graphs are presented for which |Γ′ ^ can be prepared efficiently using anapproach as described herein. 51 MatchingsA matching in a graph is a subset of edges ^ such that no two edges in ^ share acommon vertex. A perfect matching is a matching which touches every vertex of the graph.Given a set of edges ^′ = ^ of the complement graph ^^ (also referred to herein as thegraph complement) which form a matching, the state |Γ′ ^ can be prepared by configuring, on10 each pair of qubits ^, ^ such that ^^ ∈ ^, an equal superposi^on over the states00, 01, and 10, for example, by using the circuit in Figure 12 (see below).With reference to Figure 13, the edges represented by dashed lines [red] form a perfectmatching since these edges do not share any common vertex but touch every vertex of the graph. Given a matching ℳ in ^^ , we can construct the state which is a uniform15 superposi^on over states represen^ng subsets of the ver^ces of ^ which do not contain any^ edges from ℳ. The prepara^on circuit for this state has CNOT count and the CNOT depth is 1. Note that for a matching ^ that is not perfect, a single-qubit Hadamard gate isrequired on each qubit which is not touched by any edge in ^. The circuit depicted in Figure 12 may be used for crea^ng the state |Γ′ ^ =20 1 / √3(|00^ + |01^ + |10^) on 2 qubits. The gate ^^,^ sends the an^-controlled Hadamard performs a Hadamard on the second qubit when thefirst qubitreads |0^. As no two edges in a matching are incident on the same vertex, performing thiscircuit in parallel for each edge in the matching yields a uniform superposi^on over strings which do not violate ¬ any edge {^, ^} ∈ ^ (see also Figure 13 discussed above).25 Preparing the warm start state in this way allows removal of all the edges in ^ fromthe subsequent projec^on circuit. To see the savings that this makes compared with exis^ng work, consider that each edge in ^ would have u^lised a Toffoli followed by a measurementin the projec^on as shown in Figure 13. In the warm start circuit this is replaced by a one-QTDA 2 - QUK226739-WO; MJJ / 65238 PCTqubit gate and a two-qubit gate and does not require an auxiliary qubit (as per theimplementa^on of Figure 13). There are further savings due to the expected number of repe^^ons required in the projec^on-only circuit which are reduced by introducing warm-start. As an es^mate of these savings, we consider the case where the matching corresponds 5 to all edges to be removed in preparing|Γ^. In this case the probability of success for the projec^on method is (3 / 4)^ where ^ is the number of edges in the matching. This meansthe expected number of repe^^ons un^l success is achieved is (4 / 3)^which is exponen^al in ^. On the other hand, running the corresponding warm start circuit as described herein oncesucceeds with probability 1, thus requiring only a single run of the circuit. 10 2Disjoint CliquesThe quantum circuit used on each pair of qubits in the matching case may be used to prepare an equal superposi^on over all two qubit computa^onal basis states of Hamming weight 0 or 1. No^ng that an edge in ^^ is a 2-clique, we can generalise this to ^-cliques15 in ^^by construc^ng a ^-qubit circuit which prepares a uniform superposi^on over all the states |00 … 0^, |10 … 0^, |01 … 0^, … , |00 … 1^ which have Hamming weight 0 or 1. Theseare the only states which can sa^sfy^ for all ^, ^ in the ^-clique in ^. A circuit which may be used to prepare this superposi^on is given in Figure 14. In par^cular, the circuit of Figure 14 shows how to create an equal superposi^on between all ^-20 qubit computa^onal basis states containing at most 1 non-zero bit. This is precisely thestate which excludes all edges from a ^-clique. The biased Hadamard gate ^^,^sends |0^ The quantum circuit of Figure 14 may be used to generalise the construc^on from thecase of matchings. Suppose ^ = is a disjoint union of cliques in ^^; |Ψ^^ , the25 uniform superposi^on of states which exclude all the edges in ^ may be prepared by runningone of these circuits for each clique ^^in parallel. This process is depicted by way of example in Figure 15 , which shows a collec^on ofdisjoint cliques ^^, … , ^^ in ^^. In par^cular, Figure 15 illustrates two disjoint cliques basedQTDA 2 - QUK226739-WO; MJJ / 65238 PCTon the edges shown with dashed [red] line, namely afirst clique comprising nodes 1, 2 and 3,and a second clique comprising the nodes 4, 5 and 6. We can construct the state which is a uniform superposi^on over states represen^ng subsets of the ver^ces of ^ which do not contain any edges from any of the .5 The prepara^on circuit for this state has 2(^ − 1) 2-qubit gates for each ^-clique and theCNOT depth is 2(^ − 1) where ^ is the size of the largest clique in ^.The resource savings in this case are greater than in the case of using a warm start for matchings. As before, all the edges in ^ can be removed from the subsequent projec^oncircuit. Supposing this is done for a collec^on of ^ / ^ disjoint ^ cliques, this removes10 ^ / ^ × ^^^^ copies of the circuit in Figure 2 to be replaced by ^ / ^ copies of the circuit in Figure14. This involves replacing ^ / ^ × ^^^^ Toffolis with ^ / ^ × 2^ two-qubit gates, which is asaving in the order of ^ × ^ Toffolis. In terms of saving on circuit repe^^ons, the probabilityof the projec^on onto |Γ′^ succeeding when ^′ is a ^-clique is given by the propor^on of ^-bit strings containing at most 1 non-zero entries, which is (^ + 1) / 2^. For ^ / ^ disjoint ^-15 cliques, the probability of success is therefore ((^ + 1) / 2^)^ / ^, so that the number ofexpected repe^^ons un^l success is ((^ + 1) / 2^)^ / ^ . Even for rela^vely moderate values of^ and ^, this is prohibi^vely expensive for currently available quantum compu^ng hardware.3 Trees20 The above examples have involvedfinding disjoint, totally-connected cliques in ^^where there are separate circuits for preparing restricted uniform superposi^ons on each component. It will now be shown that this is not a necessary condi^on for efficientlypreparing these states since, by way of example, a recursive algorithm is provided which takes atree ^ in ^^ and constructs a quantum circuit for preparing the uniform superposi^on |Γ′^25 over states which exclude all edges in ^. This circuit construc^on method is developed induc^vely on the size of the tree ^. For the trivial one vertex tree ^^, the circuit for preparing this state is simply a single Hadamard on one qubit. Now assume that for any tree ^ of size ^ < ^, there is a circuit ^(^) on ^ QTDA 2 - QUK226739-WO; MJJ / 65238 PCT qubits in which the qubits correspond to the ver^ces of the tree and thefirst qubit represents the root of the tree such that where ^^is the number of possible assignments of 0 or 1 to each vertex in ^ which avoid 5 assigning 1 to both ends of any edge in ^ and the root is 0, and where ^^is the number of possible assignments of 0 or 1 to each vertex in ^ which respects the edge rela^on of ^ and the root is 0. The quantum circuit ^(^) ⋅ ^^^,^^ ⊗is then used to create a uniform superposi^on over all assignments ^ ∈ {0,1}^ such10 that ¬(^^ ∧ ^^) holds for each edge ^^ ∈ ^.We can now induc^vely construct the circuit ^(^′) for a tree ^′ on ^ ver^ces as follows. Let ^ be the root of ^′ and let be the subtrees rooted at the children^^, … , ^ of ^ in ^′. As each of the trees ^^ has < ^ ver^ces, by induc^on we havecircuits ^ which prepare uniform superposi^ons over15 assignments of 0’s and 1’s to the ver^ces of each subtree which exclude the edges of the trees ^^. The quantum circuit ^(^′) can be constructed by adding an an^-controlled to the qubit for ^^controlled from the qubit for ^, followed by the circuit ^(^^) on each of the subtrees. The an^-controlled gates ensure that, if 1 is assigned to ^, then the assignments to each of the subtrees range uniformly over all those assignments with 0 at each 20 of the ^^and if 0 is assigned to ^. If 0 is assigned to ^ then the assignments on the subtrees range uniformly over all permi^ed assignments to the subtrees. The circuit ^(^′) is given as follows:QTDA 2 - QUK226739-WO; MJJ / 65238 PCTTo compute ^^ and ^^, we note that if 0 is assigned to ^, then the number ofpossible assignments is given by the product of all possible assignments to the subtrees at each child of ^ - i.e. 5Similarly, if 1 is assigned to ^, we know that 0 must be assigned to each of the ^^ and sothe number of possible assignments in this case is: Figure 16 is a circuit diagram showing an example of the induc^ve step of the warm start algorithm when ^′ = ^ is a tree according to the approach described herein.10 Figure 17 provides an example of how this circuit construc^on works for a specific tree.The tree configura^on in Figure 17 is represented by the dashed [red] lines. Such a tree configura^on cannot have any loops, hence the edges from node 3 to node 4 and from node 4 to node 6 are not part of this tree. Given a set of edges ^′ which form a tree ^, we can construct the state Γ′ which is a15 uniform superposi^on over states represen^ng subsets of the ver^ces which do not contain any edges from ^. The prepara^on circuit for this state has ^ − 1 CNOT count and the CNOTdepth is the depth of the tree or the highest degree of any vertex (whichever is greater). The gate ^^,^ is a weighted Hadamard which sends |0^ to The resource saving in the case of a tree is similar to that in the case of matchings. For20 every edge in the tree, that is ^ − 1 edges for a tree of size ^, the warm start circuit uses one2 qubit gate and replaces one Toffoli and measurement block in the subsequent projec^on. The expected number of circuit repe^^ons saved can be derived from the above computa^on. For any tree, the value ^^ + ^^ computed rela^ve to the root as above is exactly the numberof 0 / 1 assignments to the ver^ces of the tree which do not assign 1 to both ends of any25 edge in the tree. Thus the probability of success of a projec^on from the equal superposi^on on all states to the state |Γ′^ defined by the tree in ques^on is (^^ + ^^) / 2^. As already seen in the example in Figure 15, this is a probability of 24 / 64, which saves, in expecta^on,roughly 3 repe^^ons of the projec^on circuit.QTDA 2 - QUK226739-WO; MJJ / 65238 PCT4 k-trees^-trees are defined induc^vely in the following way. The clique ^^ on ^ ver^ces isa ^-tree for any ^ ≤ ^. Any other ^-tree ^′ is formed by taking a ^-tree ^ and a clique^ ⊂ ^ of size at most ^ and adding a new vertex which is connected to every element of ^.5 In this terminology, 1-trees are exactly trees. An equivalent way to define a ^-tree on the ver^ces ^ is byfirst defining a tree ^ and then assigning a set of ver^ces ^^ ⊂ ^ of size atmost ^ + 1 to each node ^ ∈ ^, such that for every ^ ∈ ^, the collec^on of nodes ^ ∈ ^such that ^ ∈ ^^ forms a connected subtree. This form allows us to describe any ^-tree in acompact tree-like data structure such as shown in the example of Figure 18. In par^cular,10 Figure 18 presents an example showing a 3-tree on the le^ with its underlying tree structure on the right. Each node in the tree (right) corresponds to a clique in the network (le^). In each node of the tree, the ver^ces which do not appear in the parent underlined [green].As shown in this sec^on, the algorithm provided for a warm start over a tree can beextended to an algorithm for a warm start over a ^-tree (for any ^). The approach is to15 induc^vely describe an ^-qubit circuit ^^(^, ^) for any ^-vertex ^-tree ^ with underlyingtree ^ whose root bag ^^ has elements labelled 1, … ^ such that on input … 0^,where thefirst ^ qubits are in computa^onal basis state |^^^ = |0 =|10 … 0^, |^^^ = |01 … 0^, … , |^^^ = |0 … 01^, the output is: 20 where ^^ is the number of assignments of 0 and 1 to the ver^ces of ^ which exclude theedges of ^ and which assign the bitstring to the ver^ces 1, … ^.The tree structure of the graph allows us to compute these values for ^^recursively to construct a circuit from ^^(^, ^) which prepares the uniform superposi^on over allcomputa^onal basis states which exclude the edges of ^,first loading the weighted25superposi^on|^^^ on thefirst ^ qubits using the circuit shown in Figure 19. In par^cular, Figure 19 depicts an example of a quantum circuit for loading a weightedsuperposi^on over the ^-qubit states |00 … 0^, |10 … 0^, |01 … 0^, … |00 … 1^ with realQTDA 2 - QUK226739-WO; MJJ / 65238 PCTweights in a ra^o : … : ^^. Note that this involves ^(^) two-qubit gates. Thiscircuit can be thought of as a single weighted Hadamard ^^^ ,∑^^^ ^^on thefirst qubit, followed by a unary loader which sends the state |10 … 0^ to the appropriately weightedsuperposi^on over the Hamming weight one states. Such loaders are known from exis^ng 5work, see for example the lower depth construc^on of unary loaders due to Kerenidis &Prakash [8]. The circuit ^^(^, ^) can now be constructed by induc^on on the size of the tree ^.Suppose the root ^ of ^ has children ^^, … ^^ which are the roots of trees ^^, … ^^defining ^-trees ^^, … , which are subgraphs of ^. We have already seen in Figure 1910 that we can load a weighted superposi^on to the subsets of ^^of size at most 1, which coincide with the subsets which do not contain an edge from ^, as ^^is a clique in ^. This weight is then spread to the sub-^-trees ^^, … ^^ as follows. For each child ^^, label thever^ces ^^^ , … ^ ^^^ and the ver^ces of the intersec^on ^ ^ ^^ ∩ ^^^ as ^^ , … ^^^ . At thispoint of the circuit for ^^(^, ^), as ^^ is a clique in ^, the ver^ces in the intersec^on ^^ ∩ ^^^15 will carry a state which contains at most one 1. If any of these ver^ces carries a 1 then therest of ^^^ must be 0, which is achieved by using controlled opera^ons from each of thequbits represen^ng intersec^on ver^ces. If none of these controlled opera^ons is u^lised, the circuit should create a superposi^on over states on the remaining ver^ces of ^^^. As ^^^is a clique in the ^-tree ^, the allowed states are computa^onal basis states with at most one 1.20 These states are prepared using the same loader circuit described above in rela^on to Figure 19 (see above). We then recursively define the circuit ^^(^^ , ^^). If ^^ has a single node, thenthis circuit is the iden^ty circuit (i.e. no opera^ons). The quantum circuit for managing the above induc^ve step is shown in more detail inFigure 20. Figure 20 presents a circuit for construc^ng a uniform superposi^on over the25 subsets of ver^ces of ^^which exclude all edges in ^^given an input represen^ng a subset of ver^ces of ^^ ∩ ^^^ which contains no edges from ^. This takes an input ^ on the qubitsrepresen^ng the ver^ces in the intersec^on ^^ ∩ ^^^ . This state is assumed to be one of thecomputa^onal basis states |00 … 0^, |10 … 0^, … |00 … 1^ (or a superposi^on of these). Givena set of precomputed weights ^^ ^^ , ^^ , … ^^^^ for the subsets of with at most oneQTDA 2 - QUK226739-WO; MJJ / 65238 PCTelement and le^ng^ ^^^be the combined weight on all subsets with exactly 1 element, the gate ^^^^ ,^¬^^ followed by the block ^^^ becomes the same circuit depicted inFigure 19. The controlled opera^ons before this serve to force the state |0 … 0^ if any of theinput qubits contains a|1^. The overall result is that the qubits represen^ng the whole set 5^^^ should contain the correct distribu^on over the sets which exclude all edges in ^^^ , so thatapplying the circuit extends this to a uniform superposi^on over the subsets of ^^which exclude the edges of ^^. To complete the induc^ve construc^on over the whole ^-tree ^ this procedure isapplied to each child of ^ using separate copies of the circuit described in Figure 20 above.10 The result is the whole circuit construc^on of ^^(^, ^) shown in Figure 21. In par^cular,Figure 21 depicts an induc^ve construc^on of ^^(^, ^) for a ^-tree ^ with treedecomposi^on ^. The ver^ces {1, … ^} are the elements of the set ^^ . The children of ^are ^^, … ^^ which are roots of the subtrees ^^, … . For each ^ from 1 to ^, the setis wri^en as {^^^ , … , ^^^^^}, and the number ^^ and tuple ^^ are defined as for Figure15 20 (see above). For any ^, the circuit ^^(^^ , ^^) uses the qubits corresponding to thever^ces of - this will include some of the ver^ces in ^^ , namely those in the intersec^on^^ ∩ ^^^ .The skilled person will understand that the approach described herein may be extended to more complex graphs in terms of geometric configura^on, etc. For example, an apparatus20 may be used to prepare a quantum state represen^ng hypercliques of a D-hypergraph ^which is formed from nodes connected by hyperedges E, the hypercliques comprising sets ofnodes. Further informa^on about hypercliques can be found, inter alia, in

[0013] . The approach described herein may included the use of a classical computer system to preprocess a hypergraph complement intofirst and second subhypergraphs, thefirst subgraph comprising a 25 D-simplex k-tree of hyperedges E’ from the hypergraph complement and the second subgraph comprising remaining hyperedges E’’ from the hypergraph complement. Afirst quantumcircuit may be used to prepare afirst quantum state from the D-simplex k-tree of hyperedges (where a k-tree is a tree of edges and / or cliques, and a D simplex is a simplex having Ddimensions). A second quantum circuit may be used to prepare, based on thefirst quantumQTDA 2 - QUK226739-WO; MJJ / 65238 PCTstate and the remaining hyperedges, a second quantum state which stores insuperposi^on the hypercliques of the hypergraph. Note that such graphs may be used not only in systems havingfirst and second quantum circuits but also in systems described herein which may have just one quantum circuit, such as a circuit for warm-start or for projec^on. 5 *** The approach described herein may be u^lised across a wide range of applica^ons. One example is Quantum Topological Data Analysis (QTDA), which is a quantum machinelearning / data analysis pipeline that involves manipula^ng quantum states which represent the simplices of some simplicial complex. QTDA may be used to analyse many types of high- 10 dimensional data and may involve applica^ons of quantum circuits for topological operators such as a boundary operator. The approach described herein supports QTDA in mul^ple ways, including (i) the prepara^on of uniform superposi^on states, which is a prerequisite for certain QTDA algorithms, in a more efficient and reliable manner. The projec^on circuit described herein can also be used (ii) in the middle of QTDA algorithms such as [2] which manipulate15 quantum states such as those prepared in (i). The projec^ons prepared by the present approach allow us to restrict evolu^on to the data structure provided. Another specific applica^on for the present approach is in thefield of image processing. For example, the global topological features of images can be used to classify the images or toiden^fy features. In this situa^on, image data in 2d or 3d (with colour or density informa^on20 at each pixel / voxel) can be converted into a network (graph) given a certain scale parameter epsilon and a distance measure on the image which may combine physical distance and similarity of the addi^onal informa^on. The nodes of this network are the units of imaging data and the connec^ons (edges) represent proximity of scale epsilon between two points. Cliques on such data represent local structures in the image (lines or regions of similar 25 colour / density) and analysing how such local structures interact to form global structures (loops, voids) is an important aspect of processing image data.Accordingly, the approach described herein may be configured, inter alia, to performopera^ons such as (i) crea^ng quantum states which are superposi^ons of local structures inQTDA 2 - QUK226739-WO; MJJ / 65238 PCTimage data or (ii) using QTDA to manipulate data to project general states which represent possible local structures to the space of actual structures in an image at hand.The prepara^on of quantum superposi^ons over cliques in a graph have varied and powerful uses. The present approach replaces expensive circuits that are used to eliminate 5 ineligible cliques with more efficient quantum circuits. This is beneficial, for example, where the quantum implementa^on u^lises a sequence of hardware laser pulses, in par^cular difficult-to-execute two-ion laser pulses. The more efficient quantum circuits described herein come in two forms: circuits that avoid crea^ng easy-to-iden^fy false-start cliques to begin with, and circuits that handle mul^ple10 elimina^ons in one go. The hardware consequences of these new circuits include: fewer two- ion laser pulses and less movement of ions. The new hardware quantum circuits proposed herein will also improve the running of quantum computers for example when the CCNOT- measure primi^ve is used (even if this might not be interpreted as clique crea^on) . In addi^on, a ^^projec^on circuit such as described herein have a technical hardware 15 advantage in that they perform a rudimentary form of error "clean-up" (due to the quantum zeno effect). If errors in the form of erroneous simplices get added to a superposi^on, theprojec^on part of the quantum circuits described herein helps to eliminate these errors. Circuits that do not employ projec^on, or for which computa^on costs before a projec^onintroduces more errors than the projec^on eliminates, would not benefit from this advantage.20 This clean-up effect of projec^on is more realisable using the approach described herein, whichmay also significantly reduce total computa^onal costs. On the other hand, it may be thatgrouping too many elimina^ons into one projec^on may produce too few projec^ons to take advantage of the available quantum-zeno error clean-up effect. Accordingly, there may be a trade-off regarding the number of projec^ons to produce; this can be op^mised as appropriate. 25 Furthermore, the use of projec^on circuits ^^as described herein takes advantage of of ion-trap quantum compu^ng strengths, including iden^fying cliques of edges to eliminate ata ^me. The approach described herein allows the ions responsible for the affected ver^ces to all be moved towards the laser gate zones and to take turns to "talk" to the auxiliary qubit loaded into the gate-zone. This takes advantage of all-to-all connec^vity, in that no QuantumQTDA 2 - QUK226739-WO; MJJ / 65238 PCTSWAP circuits are needed, while not over-burdening the ion transport sub-system. In addi^on, for systems with mul^ple gate-zones, different cliques to eliminate can get sent todifferent gate-zones taking advantage of the hardware paralleliza^on capabili^es of ion-trap quantum computers. 5 The implementa^on of ^^projec^on circuits makes use of auxiliary qubits in a dynamic way to take advantage of a variable number of addi^onal qubits and the ability to reuseauxiliary qubits once they have been measured. This only works well in a se^ng where these mid-circuit measurements and resets are NOT excessively noisy (e.g. as compared to two-ion laser pulses), which is precisely the regime in which ion-trap quantum computers operate. By 10 virtue of the ability of ^^circuits to use and reuse a variable number of auxiliary qubits, this approach as described herein provides an example of circuit design that is robust to one of the known weaknesses of ion-traps, namely the loss of ions. As long as a loss occurs only on the auxiliary qubits (which is known to be detectable), the ^^circuits can recover without discarding the computa^on that has already been completed. Indeed this suggests that if for 15 example there are known differences in the life-^me of some of the ions in some of the posi^ons of the ion-trap, the more stable ions / posi^ons could be assigned to vertex qubits while the riskier ions / posi^ons could be used for the auxiliary qubits. *** In conclusion, various implementa^ons and examples have been disclosed herein. It 20 will be appreciated that these implementa^ons and examples are not intended to be exhaus^ve, and the skilled person will be aware of many poten^al varia^ons and modifica^ons of these implementa^ons and examples that fall within the scope of the present disclosure. It will also be understood that features of par^cular implementa^ons and examples can typically be incorporated into other implementa^ons and examples (unless the context clearly indicates 25 to the contrary). In summary, the various implementa^ons and examples herein are disclosed by way of illustra^on rather than limita^on, and the scope of the present inven^on is defined in the appended claims.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTThe following paragraphs provide a list of addi^onal embodiments which may serve as basis for claims in this applica^on or in any subsequentlyfiled divisional applica^on(s). Any of thefeatures of these addi^onal embodiments and other embodiments described herein may be combined with the features set out in the claims, unless stated otherwise. 5 Embodiment 1. Apparatus for performing a projec^on of a quantum state onto a subspace spanned by quantum states represen^ng (hyper)cliques which exclude a given set of hyperedges H comprising sets H2, … HD, where Hicontains (hyper)edges of size i, and wherein the apparatus is configured to: 10 use a classical compu^ng system tofind, for each 2=< i =< D, a set of (hyper)cliques Ci of size >= i in the (hyper)graph ^ whose (hyper)edges are the downward closure of those in Hi;and construct a quantum circuit onfirst and second registers wherein, for each i and each (hyper)clique K in Ci, a projec^on is performed on qubits of thefirst register onto the subspace 15 spanned by quantum states, with a count of the overlap with K less than or equal to i being stored in the second register, wherein the projec^ons in series achieve the desired projec^on onto the subspace of (hyper)cliques which exclude the (hyper)edges H. Embodiment 2. The apparatus of embodiment 1, wherein the apparatus is further configured to20 perform a series of entangling opera^ons and measurements involving the second register.Embodiment 3. The apparatus of embodiment 1 or 2, wherein the entangling opera^ons and measurements are performed without using a mul^-controlled not gate for each hyperedge. 25 Embodiment 4. The apparatus of any preceding embodiment, wherein the apparatus is configured to group the (hyper)edges together into (hyper)cliques and to perform subprojec^ons on these hyper(cliques).QTDA 2 - QUK226739-WO; MJJ / 65238 PCTEmbodiment 5. The apparatus of embodiment 4, wherein the subprojec^ons are configured to use shorter depth circuits with fewer auxiliary qubits than directly forming the desiredprojec^on. 5 Embodiment 6. The apparatus of any preceding embodiment, where the quantum circuit is configured to prepare a desired state |Γ ^ from an intermediate quantum state |Γ′ ^ .Embodiment 7. The apparatus of embodiment 6, wherein selec^on of subprojec^ons is informed by knowledge of a set ^′ of (hyper)edges which have already been removed from10 the intermediate quantum state, op^onally wherein the apparatus is configured to receive the intermediate quantum state from a warm start circuit.Embodiment 8. The apparatus of embodiment 6 or 7, where the apparatus is configured toreceive a set of remaining (hyper) edges ^′′ and wherein the quantum circuit is configured to15 separate between (i) states which contain any of these (hyper)edges E’’, and (ii) states whichcontain none of these (hyper)edges E’’.Embodiment 9. The apparatus of embodiment 8, wherein the apparatus is configured toconstruct the quantum circuit byfinding the largest, or rela^vely large, (hyper)cliques in the set20 ^′′ and arranging them into rounds of disjoint (hyper)cliques.Embodiment 10. The apparatus of embodiment 9, wherein for each disjoint (hyper)clique, thestates on the qubits represent ver^ces which can be separated.25 Embodiment 11. The apparatus of embodiment 10, wherein a subset of the ver^ces contains a(hyper)edge from the set ^′′ if and only if a set of ver^ces contains more than m elementsfrom a given subset of ver^ces, wherein computa^onal basis states |^^ … ^^^ oncorresponding qubits are accepted if and only if the Hamming weight over the given subset ofver^ces is less than m+1. 30QTDA 2 - QUK226739-WO; MJJ / 65238 PCTEmbodiment 12. The apparatus of embodiment 11, wherein iden^fying (hyper)cliques in ^′′allows a Hamming weight projec^on to form the desired projec^on. Embodiment 13. The apparatus of embodiment 11 or 12, wherein a Hamming weight projec^on5 is performed in a bitwise manner with ^(^) auxiliary qubits in two-qubit gate depth^((log^)^) and ^(log^) measurements.Embodiment 14. The apparatus of any of embodiments 6 to 13, wherein the quantum state|Γ′ ^ is the input to the desired state |Γ ^, wherein the quantum state |Γ′ ^ is not a general10 superposi^on of computa^onal basis states but rather is restricted to a subspace ℋ′ of theHilbert space qubitsEmbodiment 15. The apparatus of embodiment 14, wherein the subspace is defined by the set^′ of edges in ^^, which is a complement of (hyper)graph ^, which have been excluded from15 |Γ′ ^ by running a warm start circuit.Embodiment 16. The apparatus of embodiment 15, wherein if the warm start is in a restrictedHamming weight subspace having a maximum Hamming weight of ^′, then only ⌈log^′ + 1⌉projec^ons may be u^lised to generate the desired projec^on, thereby reducing the circuit20 depth to ^(log^ × log^′) and the number of measurements to ^(log^′).Embodiment 17. The apparatus of any of embodiments 6 to 15, wherein the apparatus isconfigured to iden^fy trivial circuit implementa^ons of a projec^on onto the set of(hyper)cliques of a (hyper)graph when the projec^on is followed by a projec^on onto the25 (hyper)cliques of a given size s.Embodiment 18. The apparatus of any preceding embodiment, wherein the apparatus isconfigured to select a set of disjoint (hyper)cliques covering as many (hyper)edges as possible for removal to create the series of projec^ons to achieve the desired projec^on, op^onallyQTDA 2 - QUK226739-WO; MJJ / 65238 PCTwherein the selec^on of a set of disjoint (hyper)cliques implements an approxima^on to covering as many (hyper)edges as possible. Embodiment 19. The apparatus of any preceding embodiment, wherein the quantum state5 represents (hyper)cliques of the (hyper)graph ^, and wherein the apparatus is configured toremove a (hyper)clique from the complement graph ^^ in each round of processing the seriesof projec^ons. Embodiment 20. The apparatus of embodiment 19, wherein the apparatus is configured to10 make a search for a set of disjoint (hyper)cliques in the whole ^^, irrespec^vely of whether some of the edges of these cliques have already been accounted for by one or more previous rounds of the projec^on. Embodiment 21. The apparatus of any preceding embodiment, wherein the apparatus is 15 configured to measure and / or manipulate the quantum state with one or more other quantum circuits to extract topological and / or geometric informa^on about the (hyper)cliques.Embodiment 22. The apparatus of any preceding embodiment, wherein the projec^on has a cost func^on which is dependent on the expected depth of the quantum circuit before afirst20 successful projec^on onto a desired subspace. Embodiment 23. The apparatus of any preceding embodiment, wherein the apparatus isconfigured to perform image processing, op^onally wherein a target (hyper)graph class is formed from image processing data. 25 Embodiment 24. The apparatus of any preceding embodiment, wherein thefirst and / or secondquantum circuits are implemented on a quantum compu^ng system comprising an ion-trapquantum compu^ng system.QTDA 2 - QUK226739-WO; MJJ / 65238 PCTEmbodiment 25. The apparatus of any preceding embodiment, wherein the apparatus is further configured to perform a separa^on of a quantum states into subspaces spanned by quantum states represen^ng (hyper)cliques which exclude different subsets of a set of hyperedges H. 5 Embodiment 26. A method for performing a projec^on of a quantum state onto a subspace spanned by quantum states represen^ng (hyper)cliques which exclude a given set of hyperedges H comprising sets H2, … HD, where Hicontains (hyper)edges of size i, and wherein the method comprises: using a classical compu^ng system tofind, for each 2=< i =< D, a set of (hyper)cliques Ci10 of size >= i in the (hyper)graph ^ whose (hyper)edges are the downward closure of those in Hi;and construc^ng a quantum circuit onfirst and second registers wherein, for each i and each (hyper)clique K in Ci, a projec^on is performed on qubits of thefirst register onto the subspace spanned by quantum states, with a count of the overlap with K less than or equal to i being 15 stored in the second register, wherein the projec^ons in series achieve the desired projec^on onto the subspace of (hyper)cliques which exclude the (hyper)edges H. Embodiment 27. The method of embodiment 26, wherein the method is implemented using the apparatus of any of embodiments 1 to 25. 20 References25 [1] “A system and method for separa^ng a quantum state into mul^ple subspaces”. GBpatent applica^on 2401344.3,filing date 1 February 2024, applicant Quan^nuum Ltd. GB2401344.3 is incorporated by reference into the present applica^on and a copy ofGB2401344.3 will be made available on thefile of the present applica^on.QTDA 2 - QUK226739-WO; MJJ / 65238 PCT[2] Ismail Yunus Akhalwaya, Shashanka Ubaru, Kenneth L. Clarkson, Mark S. Squillante,Vishnu Jejjala, Yang-Hui He, Kugendran Naidoo, Vasileios Kalantzis, and Lior Horesh. Towards quantum advantage on noisy quantum computers, 2022. [3] Israel F. Araujo, Daniel K. Park, Francesco Petruccione, and Adenilton J. da Silva. A divide-5 and-conquer algorithm for quantum state prepara^on. Scien^fic Reports, 11(1), March 2021. [4] Andreas Bartschi and Stephan Eidenbenz. Short-depth circuits for Dicke stateprepara^on. In 2022 IEEE Interna^onal Conference on Quantum Compu^ng and Engineering (QCE). IEEE, September 2022. [5] Isaac Liu Chuang and Dharmendra Shan^lal Modha. Reversible arithme^c coding for10 quantum data compression, May 2003. [6] Aric A. Hagberg, Daniel A. Schult, and Pieter J. Swart. Exploring network structure,dynamics, and func^on using networkx. In Ga¨el Varoquaux, Travis Vaught, and Jarrod Millman, editors, Proceedings of the 7th Python in Science Conference, pages 11 – 15, Pasadena, CA USA,2008.15 [7] Michael Kerber and R. Sharathkumar. Approximate ˇcech complex in low and highdimensions. In Leizhen Cai, Siu-Wing Cheng, and Tak-Wah Lam, editors, Algorithms and Computa^on, pages 666– 676, Berlin, Heidelberg, 2013. Springer Berlin Heidelberg.[8] Iordanis Kerenidis and Anupam Prakash. Quantum machine learning with subspacestates, 2022.20 [9] Seth Lloyd, Silvano Garnerone, and Paolo Zanardi. Quantum algorithms for topologicaland geometric analysis of big data. Nature Communica^ons, 082014.

[10] Sam McArdle, Andr´as Gily´en, and Mario Berta. A streamlined quantum algorithm fortopological data analysis with exponen^ally fewer qubits.92022.

[11] Fereshte Mozafari, Heinz Riener, Mathias Soeken, and Giovanni De Micheli. Effcient25 boolean methods for preparing uniform quantum states. IEEE Transac^ons on Quantum Engineering, 2:1–12, 2021.

[12] Vsevolod Salnikov, Daniele Cassese, and Renaud Lambio^e. Simplicial complexes andcomplex systems. European Journal of Physics, 40(1):014001, November 2018.

[13] W. -Z. Nie, A. -A. Liu, Y. Gao and Y. -T. Su, "Hyper-Clique Graph Matching and30 Applica^ons," in IEEE Transac^ons on Circuits and Systems for Video Technology, vol. 29, no. 6,pp.1619-1630, June 2019.

Claims

QTDA 2 - QUK226739-WO; MJJ / 65238 PCTClaims 1. Apparatus for preparing a quantum state represen^ng hypercliques of a D-hypergraph^ which is formed from nodes connected by hyperedges E whose size is between 2 and afixed5 value D, the hypercliques comprising sets of nodes whose subsets of size j between 2 and D are connected by hyperedges of size j, the apparatus configured to: generate a minimal hypergraph complement ^^; use a classical computer system to preprocess the hypergraph complement intofirst and second subhypergraphs, thefirst subgraph comprising a D-simplex k-tree of hyperedges E’ 10 from the hypergraph complement and the second subgraph comprising remaining hyperedges E’’ from the hypergraph complement; use afirst quantum circuit to prepare afirst quantum state from the D-simplex k-tree of hyperedges; and use a second quantum circuit to prepare, based on thefirst quantum state and the15 remaining hyperedges, a second quantum statewhich stores in superposi^on thehypercliques of the hypergraph.

2. Apparatus for preparing a quantum state represen^ng cliques of a graph ^ which isformed from nodes connected by edges E, the cliques of the graph comprising nodes that are20 all pairwise connected by edges, the apparatus configured to: generate a graph complement ^^of the graph; use a classical computer system to preprocess the graph complement intofirst and second subgraphs, thefirst subgraph comprising a k-tree of edges E’ from the graph complement and the second subgraph comprising remaining edges E’’ from the graph 25 complement; use afirst quantum circuit to prepare afirst quantum state from the k-tree of edges; andQTDA 2 - QUK226739-WO; MJJ / 65238 PCTuse a second quantum circuit to prepare, based on thefirst quantum state and the remaining edges, a second quantum statewhich stores in superposi^on the cliques of thegraph.

53. The apparatus of claim 1 or 2, wherein the (hyper)edges E’ of thefirst subgraph aredisjoint with respect to the (hyper)edges E’’ of the second subgraph.

4. The apparatus of any preceding claim, wherein the second quantum circuit is configuredto prepare a uniform superposi^on over the (hyper)cliques of a given (hyper)graph.10 5. The apparatus of any preceding claim, wherein the second quantum circuit uses aprojec^on to prepare the second quantum state, op^onally wherein the second quantum circuit is configured to use a Hamming weight projec^on to prepare the second quantum state.15 6. The apparatus of any preceding claim, wherein thefirst quantum circuit is adeterminis^c state prepara^on circuit, wherein the determinis^c state prepara^on circuit usesone or more classes of (hyper)edges E’ from thefirst subgraph to prepare thefirst quantum state.20 7. The apparatus of any preceding claim, wherein thefirst quantum circuit comprises awarm start circuit which is configured to process the set of (hyper)edges E’ to create determinis^cally thefirst quantum state |Γ^^, where |Γ^^ is a superposi^on over all cliqueswhich do not contain edges from ^′.25 8. The apparatus of any preceding claim, wherein thefirst quantum state corresponds to aVietoris-Rips complex defined on the graph ^′ which has the same ver^ces as ^ but hasedges defined by the rela^on ^((^′)^) = ^′.QTDA 2 - QUK226739-WO; MJJ / 65238 PCT9. The apparatus of any preceding claim, wherein the preprocessing prepared by theclassical computer system comprises genera^ng viable decomposi^ons of the set ^(^^) intosets ^′ and ^′′ using one or more classical methods and then choosing a decomposi^on,from the viable decomposi^ons, which minimises a cost func^on ^(^′, ^′′).5 10. The apparatus of any preceding claim, wherein for a set of (hyper)edges ^, theapparatus is configured tofind a largest set ^′ which forms a graph from a target class,op^onally wherein the apparatus is configured to use classical processing tofind the largest set E’. 10 11. The apparatus of any preceding claim, wherein the second quantum circuit is furtherconfigured to perform a projec^on on a warm start state prepared by thefirst quantum circuit to remove all (hyper)edges from ^\^′ to form a desired state |Γ ^.15 12. The apparatus of any preceding claim, wherein the apparatus is configured to use atarget (hyper)graph class to provide an efficient state prepara^on circuit, wherein the target(hyper)graph class comprises one of: • Matchings • Sets of disjoint cliques 20 • Trees • ^-trees which are trees of cliques • D-simplex k-trees.

13. The apparatus of any preceding claim, wherein the k-tree of (hyper)edges E’ comprises25 all of the (hyper)edges from the (hyper)graph complement such that there are no remaining (hyper)edges, whereby thefirst quantum state directly represents the second quantum stateQTDA 2 - QUK226739-WO; MJJ / 65238 PCT14. The apparatus of any preceding claim , wherein thefirst quantum circuit stores insuperposi^on the (hyper)cliques of the (hyper)graph without using the second quantum circuit.

15. The apparatus of any preceding claim, wherein the k-tree is used to prepare thefirst5 quantum state |Γ^^ which is a uniform superposi^on over all quantum states represen^ngsubsets of the ver^ces which do not contain any edges from ^′, wherein ^′ has the samever^ces as ^ but has edges defined by the rela^on= ^′ , wherein thefirst quantumcircuit has a structure which follows the k-tree, and wherein each node is configured to generate a quantum subcircuit which excludes all edges in the clique represented by the node.10 16. The apparatus of any preceding claim, wherein for a desired state restricted to asuperposi^on of hypercliques of afixed size, the apparatus is configured to perform addi^onalclassical preprocessing to iden^fy ver^ces which are not involved in any (hyper)cliques of size kand to adapt thefirst and second quantum circuits to reduce qubit count and / or circuit depth.15 17. Apparatus for preparing a warm start quantum state represen^ng hypercliques of a D-hypergraph ^ which is formed from nodes connected by hyperedges E whose size is between2 and afixed value D, the hypercliques comprising sets of nodes whose subsets of size between 2 and D are connected by hyperedges of size D, the apparatus configured to: 20 generate a minimal hypergraph complement ^^; use a classical computer system to process the hypergraph complement into a subgraph comprising a D-simplex k-tree of hyperedges E’ from the hypergraph complement; anduse a quantum circuit to prepare the quantum state from the D-simplex k-tree ofhyperedges. 25 18. Apparatus for preparing a warm start quantum state represen^ng cliques of a graph ^which is formed from nodes connected by edges E, the cliques of the graph comprising nodesthat are all pairwise connected by edges, the apparatus configured to: generate a graph complement ^^of the graph;QTDA 2 - QUK226739-WO; MJJ / 65238 PCTuse a classical computer system to preprocess the graph complement into a subgraph comprising a k-tree of edges E’ from the graph complement; use a quantum circuit to prepare the warm start quantum state from the k-tree ofedges. 5 19. The apparatus of claim 17 or 18, wherein the quantum circuit is configured to prepare auniform superposi^on over computa^onal basis states which do not contain a (hyper)edge of a ^-tree.10 20. The apparatus of any of claims 17 to 19, wherein the apparatus includes at least oneclassical rou^ne configured to generate mul^ple viable decomposi^ons of the set E(^^) and to choose one of the mul^ple viable decomposi^ons which minimises a cost func^on based on the circuit depth of the quantum circuit.15 21. The apparatus of any of claims 17 to 20, wherein the quantum circuit is configured toproduce a uniform quantum state |Γ′ ^ which can be projected onto a desired quantum state|Γ ^, wherein the quantum circuit has a shorter depth and / or a higher probability of successthan a quantum circuit that would otherwise be used to project to |Γ ^ from a uniformsuperposi^on |Ψ ^.20 22. The apparatus of any of claims 17 to 21, wherein the classical computer system isconfigured to perform a preprocessing rou^ne to split the complement (hyper)graph ^^into two edge-disjoint subgraphs, such that each (hyper)edge from complement (hyper)graph ^^is in one but not both of the two subgraphs. 25 23. The apparatus of any of claims 17 to 22, wherein based on the set of edges E', thequantum circuit is configured to create a quantum state |Γ′ ^ corresponding to a Vietoris-Ripscomplex defined on a graph G' which has the same ver^ces as G but has edges defined by therela^on = ^′, wherein E' is the set of edges of the complement graph of G'.QTDA 2 - QUK226739-WO; MJJ / 65238 PCT24. The apparatus of claim 23, wherein the(hyper) edges ^′ are selected so that the(hyper)graph has a structure amenable to a warm start for preparing the quantum state, wherein the classical computer system performs pre-processing to iden^fy related structures in 5 the complement (hyper)graph ^^.

25. The apparatus of claim 24, wherein said related data structures include one or more ofmatchings, sets of disjoint cliques, trees, and ^-trees etc. which are trees of cliques.10 26. The apparatus of any of claims 17 to 25, wherein different target graph classes areu^lised to produce a uniform quantum state |Γ′ ^, the graph classes including:• Matchings • Sets of disjoint cliques • Trees 15 • ^-trees which are trees of cliques. • D-simplex k-trees.

27. The apparatus of any preceding claim, wherein thefirst quantum circuit is performeddeterminis^cally with no quantum measurements. 20 28. The apparatus of any preceding claim, wherein the apparatus is configured to performimage processing, op^onally wherein a target (hyper)graph class is formed from image processing data.25 29. The apparatus of any preceding claim, wherein thefirst and / or second quantum circuitsare implemented on a quantum compu^ng system comprising an ion-trap quantum compu^ngsystem.QTDA 2 - QUK226739-WO; MJJ / 65238 PCT30. A method for preparing a quantum state represen^ng hypercliques of a D-hypergraph^ which is formed from nodes connected by hyperedges E whose size is between 2 and afixedvalue D, the hypercliques comprising sets of nodes whose subsets of size between 2 and D are connected by hyperedges of size D, the method comprising:5 genera^ng a minimal graph complement ^^ of the graph;using a classical computer system to preprocess the hypergraph complement intofirstand second subhypergraphs, thefirst subgraph comprising a k-tree of hyperedges E’ from the hypergraph complement and the second subgraph comprising remaining hyperedges E’’ from the hypergraph complement;10 using afirst quantum circuit to prepare afirst quantum state from the k-tree of edges;and using a second quantum circuit to prepare, based on thefirst quantum state and theremaining hyperedges, a second quantum statewhich stores in superposi^on thehypercliques of the hypergraph. 15 31. A method for preparing a quantum state represen^ng cliques of a graph ^ which isformed from nodes connected by edges E, the cliques of the graph comprising nodes that areall pairwise connected by edges, the method comprising:genera^ng a graph complement ^^ of the graph;20 using a classical computer system to preprocess the graph complement intofirst andsecond subgraphs, thefirst subgraph comprising a k-tree of edges E’ from the graph complement and the second subgraph comprising remaining edges E’’ from the graph complement; using afirst quantum circuit to prepare afirst quantum state from the k-tree of edges;25 and using a second quantum circuit to prepare, based on thefirst quantum state and theremaining edges, a second quantum statewhich stores in superposi^on the cliques of thegraph.QTDA 2 - QUK226739-WO; MJJ / 65238 PCT32. A method for preparing a quantum state represen^ng hypercliques of a D-hypergraph^ which is formed from nodes connected by hyperedges E whose size is between 2 and afixedvalue D, the hypercliques comprising sets of nodes whose subsets of size between 2 and D are connected by hyperedges of size D, the method comprising:5 genera^ng a minimal hypergraph complement ^^;using a classical computer system to process the hypergraph complement into asubgraph comprising a D-simplex k-tree of hyperedges E’ from the hypergraph complement; and using a quantum circuit to prepare the quantum state from the D-simplex k-tree of10 hyperedges.

33. A method for preparing a warm start quantum state represen^ng cliques of a graph ^which is formed from nodes connected by edges E, the cliques of the graph comprising nodesthat are all pairwise connected by edges, the method comprising:15 genera^ng a graph complement ^^ of the graph;using a classical computer system to preprocess the graph complement into a subgraphcomprising a k-tree of edges E’ from the graph complement; using a quantum circuit to prepare the warm start quantum state from the k-tree ofedges. 20 34. The method of any of claim 30 to 33, wherein the method is implemented using theapparatus of any of claims 1 to 29. 25

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