Posner computing: a quantum neural network model in Quipper 6 James L. Ulrich∗ 1 0 January 2, 2017 2 c e D Abstract 0 We present a construction, rendered in Quipper, of a quantum algorithm which 3 probabilisticallycomputesaclassicalfunctionfromnbitstonbits. Theconstructionis intendedtobeofinterestprimarilyforthefeaturesofQuipperithighlights. However, ] h intrigued by the utility of quantum information processing in the context of neural p networks, we present the algorithm as a simplest example of a particular quantum - neural network which we first define. As the definition is inspired by recent work of t n Fisherconcerningpossiblequantumsubstratestocognition,weprecedeitwithashort a description of that work. u q [ Acknowledgement 2 v TheauthorwishestothankNeilJ.Rossforhelpfuldiscussionsandcodecontributions, 7 and Paul Li, Mark Raugas, and Riva Borbely for helpful reviewing. 3 1 1 Introduction 7 0 . We seek in this work to add to the corpus of quantum algorithms expressed in the 1 Quipper quantum programming language [7], by presenting a construction of a quan- 0 tum algorithm which probabilistically computes a classical function from n bits to n 6 1 bits. Of course, since a general quantum circuit can compute exactly any algorithm : constructed using classical gates [8], the algorithm is intended to be of interested pri- v marily for the features of Quipper it highlights, and for its restriction to particular i X types of gates. The code itself is presented in the appendix. However, intrigued by the utility of quantum information processing in the context of neural networks, we r a presentitasasimplestconstructionofaparticularquantumneuralnetworkwhichwe define below. Thedefinition,inturn,isinspiredbyanarticlebyFisher[4],inwhichheconjectures that the phosphate ion’s half-integer spin may serve as the brain’s “qubit” (i.e., unit of quantum storage), and that pairs of such ions form spin singlet states, which are preserved inside “Posner molecules”. These cube-shaped molecules inherit a tri-level “pseudo-spin”fromthesixphosphateionstheycontain,characterizingspineigenstate transformations under rotations along the cube diagonal. Posner molecules may bond in pairs, collapsing onto a zero total pseudo-spin state leading to release of calcium, ∗Staffscientist,CyberPointInternationalLLC;email: jlu5@caa.columbia.edu. 1 Posner computing: a quantum neural network model in Quipper 2 which in turn enhances neuronal firing. Posner molecules in different neurons are posited to become entangled, producing cross-neuronal firing correlations which are quantum in origin. Hence, we consider a model for a feed-forward, layered neural network inspired by the described interactions of Posner molecules. The network resembles a traditional feed-forward neural network, except that in any given layer, the activation functions of one or more of the units are replaced with a single quantum operation, using a specified (not necessarily universal) gate set, and taking the units’ classical inputs to their classical outputs. The algorithm presented below may be seen as a simplest example of such a network. 2 Related work From a philosophical perspective, there is a grand tradition of attempting to model cognition as an artifact of quantum information processing in the brain. In his 1989 book “The Emperor’s New Mind,” Penrose introduced the idea that consciousness is anartifactofthegravity-inducedcollapseofaquantum-mechanicalwave-functiongov- erningbrainstates,referredtoas“OrchestratedObjectiveReduction(Orch-Or)”[12]. A few years later, Albert [1] explored the non-intuitive consequences for mental belief states of the application of the Copenhagen interpretation of quantum mechanics to humanobservers[1],suggestingthestandardinterpretationiswrong,therebysupport- ing Penrose. In the following decade, work by Khrennikov [10] posited that the brain is a “quantum-like” computer, in that one may observe interference patterns in sta- tistical descriptions of mental states. An updated version of the Penrose’s “Orch-Or” hypothesisisprovidedin[9],andFisher’srecentarticle[4]mightreasonablybeviewed as providing an alternative physical substrate for this hypothesis. A key element of this substrate is that while the processes describing neuronal inputs and outputs may be classical, groups of neurons may coordinate firing through quantum processes. It isthiselementwhichmotivatesthedescriptionoftheneuralnetworkdescribedbelow, which we dub a “Posner” network, in honor of its origin. In terms of extending classical neural networks to the quantum world, approaches maybefoundin(forexample)[13]and[2],andasurveymaybefoundin[6]. Quantum perceptron networks and/or perceptron networks using quantum computation in the training phase are given in (for example) [20], [21], and [22], and a review of some approaches may be found in [17]. A general framework for quantum machine learning is presented in [3]. In the model presented here, quantum states exist only within a network layer; the layer inputs and outputs are purely classical. Moreover, we stress that our primary aim is simply to contribute to the corpus of extant Quipper code. 3 Posner molecule interactions For the interested reader, we here provide a brief summary of the aspects of [4] mo- tivating the present work, and encourage the reader to consult that source for more detail. The remainder of this paper does not depend on this section, however. The mechanism for quantum cognition described in Fisher’s work proceeds more or less as follows. It starts with two phosphate ions with nuclear spin 1 held inside a 2 magnesium shell in extracellular fluid, and emitted in a spin singlet state 1(|0(cid:105)−|1(cid:105)). 2 The entangled ions are then drawn into (possibly distinct) presynaptic glutematergic neurons, wheretheyparticipateintheformationofPosnermolecules, eachcontaining six phosphorous ions. The ions, when viewed along the diagonal 3-fold symmetry axis of the molecule, form a hexagonal ring, and the associated spin Hamiltonian is given Posner computing: a quantum neural network model in Quipper 3 by H = (cid:80) J S(cid:126) ·S(cid:126) where i,j = 1,2,··· ,6 label the 6 spins, the S(cid:126) are the spin ij ij i j l operators,andtheJ areexchangeinteractions. The26energyeigenstatesofH canbe ij labelled by their transformation properties under a 3-fold rotation, acquiring a phase factoreiσ2π/3 withσ=0,+1,−1.Hencethemoleculecanbedescribedbypseudo-spin (cid:80) states|0(cid:105),|1(cid:105),|−1(cid:105),withoverallmolecularstateoftheform|Ψ (cid:105)= c |ψ (φ)(cid:105)|σ(cid:105), Pos σ σ σ whereψ encodesrotationalspinstateforangleφaboutthediagonal, giventhestate σ |σ(cid:105). [5]. The state of two Posner molecules (a,a(cid:48)) occupying a neuron is given by |Ψaa(cid:48)(cid:105)=(cid:88)Cσaσa(cid:48)(cid:48)|ψσa(cid:105)|ψσa(cid:48)(cid:48)(cid:105)|σσ(cid:48)(cid:105)aa(cid:48), (cid:88)|Cσaσa(cid:48)(cid:48)|2 =1. σ,σ(cid:48) σ,σ(cid:48) which is entangled unless Caa(cid:48) =caca(cid:48), and where by ca (respectively, ca(cid:48)), we mean σσ(cid:48) σ σ(cid:48) σ σ(cid:48) the coefficient of the σ component of the pseudo-spin of molecule a (respectively, σ(cid:48) component of a(cid:48)). Chemical bonding of the molecules is equivalent to collapse of |Ψaa(cid:48)(cid:105) onto a total pseudo-spin 0 state. In this case both molecules melt, releasing calcium, which in turn stimulates further glutamate release into the synaptic cleft, impacting the firing behavior of the post-synaptic neuron. The probability of bonding is given by Praeaa(cid:48)ct =(cid:88)|Cσaσa(cid:48)(cid:48)|2δσ+σ(cid:48),0, (1) σ,σ(cid:48) where δ denotes the Kronecker delta function. Now let |Ψaa(cid:48)(cid:105)⊗|Ψbb(cid:48)(cid:105) be the state encoding two pairs of Posner molecules (a,a(cid:48)) and (b,b(cid:48)) with (a,b) in neuron 1, and (a(cid:48),b(cid:48)) in neuron 2, with (a,a(cid:48)) entangled, as are (b,b(cid:48)). Let r =1 if a,b bind and 0 otherwise, and similarly let r(cid:48) =1 if a(cid:48),b(cid:48) bind and 0 otherwise. Then the joint probability of a given combination r,r(cid:48) is given by Prr(cid:48) = (cid:88) (cid:88) |Cσaaaσ(cid:48)a(cid:48)|2|Cσbbb(cid:48)σb(cid:48)|2gr(σa,σb)gr(cid:48)(σa(cid:48),σb(cid:48)) (2) σaσa(cid:48)σbσb(cid:48) where g1(σ,σ(cid:48))=δσ+σ(cid:48),0 and g0 =1−g1. Fisher defines an “entanglement measure” E = [δrδr(cid:48)] where δr := r −[r] and [frr(cid:48)] := (cid:80)rr(cid:48)Prr(cid:48)frr(cid:48). If E > 0 then the two binding reactions themselves are pos- itively correlated by virtue of quantum entanglement, and if E < 0 then they are anti-correlated. We seek to capture this feature of interneuronal entanglement in our definition of a “Posner” network, which follows in the next section. 4 A quantum neural network for Posner computing We now define a quantum neural network inspired by the elements of the preceding section. For convenience and to fix notation, we recall that a feed-forward, hard- threshold perceptron network [15], may be thought of as a map from Zn →Zm, given 2 2 by a direct acyclic graph G with vertices v , i=1,··· ,n , organized into layers L , ij Lj j j =1,···k, with edges eij,i(cid:48)(j+1) running only between layers Lj and Lj+1. For every layer Lj aside from the first, each vi(cid:48)j takes a value given by a binary-valued function gi(cid:48)j((cid:80)wi(j−1),i(cid:48)jvi(j−1)+d(cid:48)i),wherethesumistakenoverallvi inlayerLj−1 suchthat there is an edge ei(j−1),i(cid:48)j from vi(j−1) to vi(cid:48)j, and the wi(j−1),i(cid:48)j are weights assigned to the respective edges, and d is a constant. To evaluate the map on a binary input i b ,b ,··· ,b , we set the v =b . Iteratively evaluating each successive layer in turn, 1 2 n i1 i Posner computing: a quantum neural network model in Quipper 4 Figure 1: A two-layer SPN-R. The outputs of units two and three inlayer2aredeterminedbyaunitarytransformationonastatecor- respondingtothevaluesofunitstwoandthreeinlayerone,followed by a projective measurement. given the inputs provided by the preceding layer, the corresponding output will then be given by the values of the units v ,··· ,v in the last layer L . To obtain our 1k mk k quantumneuralnetwork,wemodifythisclassicalneuralnetworkinthefollowingway: Definition4.1. ASimplePosnerR-Network (SPN-R)isafeed-forward,hard-threshold perceptron network as described above, with the following amendations. For at least some units v ,··· ,v in at least one layer L : i1j ipj j 1. The functions g are replaced with a single quantum circuit [8] on p qubits, ij consisting of a single unitary transformation U = U : C2⊗p → C2⊗p, (i1,···,ip),j followedbyaprojectivemeasurementontoacomputationalbasisstate. Thepos- sible values of v ,··· ,v form the 2p computational basis vectors b of C2⊗p. i1j ipj q ThecircuitU mustbeconstructedfromaspecific(notnecessarilyuniversal)gate set R. 2. Evaluating U on a given basis vector b produces a vector v = (cid:80)2p c b . Ran- q=1 q q domlyprojectingontoabasisvectorb ,whereb ischosenwithprobability|c |2, q q q determines the outputs of the v ,··· ,v , where the r-th bit of b is the value i1j ipj q assigned to v . rj 3. Eachvij ∈{vi1j,··· ,vipj}isconnectedtoexactlyoneunitvi(cid:48)(j−1) inlayerLj−1, andtheconnectingedgehasunitweight. ThebasisvectoronwhichU isevaluated is determined by the values of the vi(cid:48)(j−1). An example of an SPN-R appears in figure 1. It follows from definition 4.1 that a quantum circuit implementing a given unitary transformation U on a quantum register initialized from n classical bits, followed by a Posner computing: a quantum neural network model in Quipper 5 Figure 2: A three-layer SPN-R consisting of a Hadamard transformation (in L ) on a singe qubit initialized to the output of layer 2 L ,followedaprojectivemeasurement,outputofwhichpassestoL , 1 3 whereanotherHadamardtransformisperformed,followedbyanother measurement. The SPN-R does not correspond to a single unitary transformation, because of the projective measurement in layer L . 2 projectivemeasurementontoann-bitbasisstatefromwhichann-bitclassicaloutputis read,maybeviewedasatwo-layerSPN-R. More(perhaps)interestingconstructions, in terms of the Quipper implementations to which they lead, arise from restricting the gate set R, or combining such two-layer SPN-R . Clearly there are SPN-Rs that cannotberepresentedbyasingleunitarytransformation: forexample,thethree-layer SPN-R coupling two copies of a single-qubit quantum circuit performing a Hadamard transformationfollowedbyaprojectivemeasurementinthe|0(cid:105),|1(cid:105)basis(seefigure2). 4.1 SPN-Rs can learn arbitrary classical functions Toillustratethecapability(andlimitations)ofthesimplestSPN-Rconfiguration,and tomotivatetheconstructionofourQuippercircuit,wepresentthefollowingtheorem. Theorem 4.1. Let n be an integer such that there exists an angle θ∈(0,π) such that 2 2n−2 √1 (cid:0)cos2n−1(θ)+ (cid:88) cosi(θ)sin(θ)(cid:1)> √1 . (3) n 2 2 i=0 LetRbethequantumgatesetconsistingoftheHadamardgate,swapgate,andcon- (cid:18) (cid:19) cos(θ) sin(θ) trolled,two-levelgateswhichactonbasisstates|0(cid:105),|i(cid:105)andarebuiltfrom . sin(θ) −cos(θ) Thenforanysurjectivefunctionf :Zn →Zn takingn-bitbinarystringston-bitbinary 2 2 strings, there is a two-layer SPN −R which can “learn” f, in that given an input b, it computes f(b) with probability greater than 1. 2 Proof. We construct the SPN-R in following way. A surjective function f with range Zn takes 2n distinct values, corresponding to the basis vectors |i(cid:105) of C2⊗n. Taking 2 |0(cid:105) ∈ C2⊗n and applying a Hadamard transform yields an even superposition |ψ(cid:105) of all basis vectors. To bias this state towards a given basis vector |k(cid:105) corresponding to f = k, one can apply, for each j, a y-axis rotation of fixed angle θ towards |k(cid:105) in the Blochspherecorrespondingto{|k(cid:105),|j(cid:105)}. Foraprojectivemeasurementon|ψ(cid:105)toselect |k(cid:105) with odds greater than 1, θ must satisfy (3); the left-hand side of the inequality 2 Posner computing: a quantum neural network model in Quipper 6 Figure3: Quantumcircuitcomputingthe“complement”functionforthetwoqubit case. representstheoveralltransformationoftheco-efficientof|k(cid:105)withinthebasisexpansion of |ψ(cid:105). Couchedintermsof“learning”, wetakethetrainingdatatobethesetof2n pairs of n-digit binary strings {b ,b } defining f, where b is an input to f and input output input b is the associated output. The SPN-R takes the form of a quantum circuit output acceptingnclassicalbits,onwhichitperformsasingle(multi-gate)unitaryoperation, followed by a projective measurement onto the computational basis. The quantum circuit consists of n quantum input bits (qubits) and n quantum ancilla qubits. The circuit first initializes n ancilla qbits to the |0(cid:105) state, and then performs a Hadamard (cid:18) (cid:19) transformation to take |0(cid:105) → ψ1 = √12n (cid:80)2i=n0−1|i(cid:105). For U = scoins((θθ)) −scinos((θθ)) , and each 0 < i ≤ 2n−1, a two-level unitary transformation U is performed on the 0,i spacespannedbythebasisstates|0(cid:105)and|i(cid:105)(U actstriviallyonallothersubspaces). 0,i Collectively, the U take ψ to a state ψ in which the |0(cid:105) component has amplitude 0,i 1 2 √12n(cid:0)cos2n−1(θ)+(cid:80)2i=n0−2cosi(θ)sin(θ)(cid:1). Then, for each boutput, the circuit performs a controlled, two-level X gate swapping |0(cid:105) and |b (cid:105), the latter being the basis output statewhosei-thqubitcorrespondstothevalueofthei-thbitofb ,andwherethe output controlsaregivenbyb andplacedontheinputqubits. Hencetheamplitudeof|0(cid:105) input and |b (cid:105) are swapped precisely when the input qubits are initialized according to output b . input AQuipper-basedimplementationoftheabovealgorithmmaybefoundat[18]and is reproduced in the appendix. The implementation uses θ= π for even n, and 2(n/2)+1 θ = π for odd n. These choices of θ have been computer-verified to satisfy 2((n−1)/2)+1 (3) for 1 the first 2ˆn lines define a bijective function from bitstrings of length n to bitstrings of length n; the digits to the left of the ’:’ are the inputs. the lines must appear in increasing order of outputs , viewed as binary strings (e.g. 00, 01, 10, 11, for n=4). the next to last line is a specific input to be evaluated by simulating the circuit (used for mode ”S” only). the last line is a run mode: A=show amplitudes via sim amps (supported for n <= 2 only), for input |0>. P=print generic. S=simulate via run generic. −− major algorithmic steps (in concept , if not actual code) are: 1. ingest a string giving the map of (input ,output) pairs and store in a suitable data structure in memory. 2. define an n−quibit circuit that takes the |0> state to one that is a superposition S of all possible n−qubit basis states , but for which the state |0> has the largest probability amplitude. We do by first applying a hadamard gate , and then applying the two level ”bias” gate [[cos(pi/f(n)), sin(pi/f(n))], [sin(pi/f(n)), −cos(pi/f(n))]] successively to states (|0>,|1>), (|0>,|2>), .... , (|0>,|2ˆ(n−1)>), for suitable f(n). 3. add n ancilla qubits to the circuit. then for each output i given by the map, add a controlled swap gate exchanging the weight of the |0> component of S with that of the component corresponding to |i>, with controls given by the input producing |i>. the controls are placed on the ancillas. 4. to evaluate the circuit for a given input , the ancillas are initialized to the input. −} −− declare a map data type entry data InputOutput = InputOutput { input :: [Bool], output :: [Bool] Posner computing: a quantum neural network model in Quipper 9 } deriving (Show) −− declare a function truth table (input−>output map defining a function) data TruthTable = TruthTable { entries :: [InputOutput] } deriving (Show) −− produce a quantum circuit from the function defined by the input processInput :: [String] −> ([Qubit] −> Circ [Qubit]) processInput table = do −− from input lines defining a function (as a text−based truth table of −− inputs−>outputs), get list of input ,output string pairs let u = map (\x −> splitOn ”:” x) table −− from list of input , output string pairs , get list of −− input , output boolean list pairs , make internal truth table from that let v2 = map (\x −> (str to blist(x !! 0),str to blist(x !! 1)) ) u let g2 = map (\x −> InputOutput (fst x) (snd x)) v2 let tt = TruthTable g2 −− make quantum circuit based on table let circ = tt circuit tt circ −− convert a text string of 0s and 1s to a boolean array str to blist :: String −> [Bool] str to blist lc = let bl = map (\x −> case x of ’1’ −> True ’0’ −> False) lc in bl −− A basis change to obtain y−rotations from z−rotations. −− courtesy Neil J. Ross. y to Z :: Qubit −> Circ Qubit y to Z q = do q <− gate S inv q q <− gate H q q <− gate S inv q return q {− the ”bias” gate R Y(pi/f(n))∗Z; see Nielsen and Chuang, Quantum Computation and Quantum Information , pp.175−176. Decomposition in terms of Quipper gates courtesy Neil J. Ross. −} my biasUt :: Timestep −> Qubit −> Circ () my biasUt t q = do Posner computing: a quantum neural network model in Quipper 10 −− q <− expZt pi q q <− gate Z q q <− with computed (y to Z q) (expZt t) return () apply controlled X :: ControlList −> Qubit −> Circ() apply controlled X controls q = do q <− gate X q ‘controlled ‘ controls return () −− repeatedly apply our two−level bias gate as −− a U {0,i} gate for i , i−1, i−2, ... , 1 bias gate recurse :: (Qubit−> Circ()) −> Int −> [Qubit] −> Circ() bias gate recurse g 0 qbits = return() bias gate recurse g i qbits = do twolevel 0 i qbits g bias gate recurse g (i−1) qbits −− construct a control list for a given set of function inputs , expressed −− as bools control recurse :: ControlList −> Int −> [Qubit] −> [Bool] −> ControlList control recurse controls idx qubits bools = case (idx == 0) of True −> controls False −> do let new controls = controls .&&. (qubits !! idx) .==. (bools !! idx) control recurse new controls (idx−1) qubits bools −− apply a controlled X gate as U {0,idx} with controls on inputs corresponding −− to output state |idx> for thruth table entries idx = 1....n flip gate recurse :: TruthTable −> Int −> [Qubit] −> [Qubit] −> Circ() flip gate recurse tt idx ancillas qbits = do −− get the input booleans for the output given by idx let pos neg = input ( (entries tt) !! idx ) −− now build a control list based on the booleans. we recurse to build it −− up wire by wire using the .&&. operator let init controls = (ancillas !! 0) .==. (pos neg !! 0) let controls = control recurse init controls (length(pos neg) −1 ) ancillas pos neg let g = apply controlled X controls twolevel 0 idx qbits g case (idx == (length (entries tt) − 1)) of True −> return() False −> flip gate recurse tt (idx + 1) ancillas qbits

Posner computing: a quantum neural network model

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