A geometrical approach to quantum holonomic computing algorithms

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Abstract

The article continues a presentation of modern quantum mathematics backgrounds started in [Quantum Mathematics and its Applications. Part 1. Automatyka, vol. 6, AGH Publisher, Krakow, 2002, No. 1, pp. 234–2412; Quantum Mathematics: Holonomic Computing Algorithms and Their Applications. Part 2. Automatyka, vol. 7, No. 1, 2004]. A general approach to quantum holonomic computing based on geometric Lie-algebraic structures on Grassmann manifolds and related with them Lax type flows is proposed. Making use of the differential geometric techniques like momentum mapping reduction, central extension and connection theory on Stiefel bundles it is shown that the associated holonomy groups properly realizing quantum computations can be effectively found concerning diverse practical problems. Two examples demonstrating two-form curvature calculations important for describing the corresponding holonomy Lie algebra are presented in detail.

Introduction

Consider a two-dimensional vector subspace F(2)H of some complex Hilbert space H called further a quantum bit (qubit) information space as follows: F(2):=spanC{|0>,|1>∈H}with orthonormal base vectors |0>,|1> H subject to the scalar product 〈·,·〉 in H and also its nth tensor product H(n):=F(2)⊗Fn-times(2)⊗⋯⊗F(2)being a Hilbert space of dim H(n)=2n, called usually an n-qubit computation medium. Now we will implement as one of the simplest computational tasks in this medium, any classical polynomially computable function f:RR: that is we will calculate all values f(x),x∈R, at the same time (with needed accuracy).

Take any x∈R and represent it in two-digital {0,1}-system, thereby we can write down the bijection x↔(x1,x2,x3,…,xn)2,where by definition x=∑i∈Zxi2i,xi∈{0,1}for any i∈Z. Such a sequence (x1,x2,x3,…,xn)2 can now be bijective imbedded into the above constructed Hilbert space H(n) of n-qubit states: R∋x↔(x1,x2,x3,…,xn)2↔|x1,x2,x3,…,xn>∈H(n).So, for any x∈R one can construct an n-qubit state via the induced mapping R∋x↔|x1,x2,x3,…,xn>:=|(x)>∈H(n).On the set {0,1}∈Z2 one has the usual algebraic operations: if x,y∈Z2, then x⊕y:=x+y(mod2),or 0⊕0=0,1⊕1=0,0⊕1=1,1⊕0=1,0·1=0,1·1=1,1·0=0.

Definition 1

The following unitary transform of the Hilbert space H(n+k) for a mapping f:Z2nZ2k,n,k∈Z+, Uf:|(x),(a)>↦|(x),(a)(f(x))>,where values (f(x)) and (a) are meant correspondingly as (f1(x),f2(x),…,fk(x))2 and (a1,a2,…,ak)2 with respect to the mapping (6), is called a quantum computation subject to any fixed test vector |(a)>∈H(k),k∈Z+.

This computation is usually exposed [22] graphically as From (9) and due to linearity one gets easily Uf:∑x∈Z+αx|(x),(a)>↦∑x∈Z+αx|(x),(a)(f(x))>for all αxC,x∈Z+. Thus, one sees that the unitary operator Uf:H(n+k)H(n+k) counts all values of pairs |(x),(f(x))>,x∈R, which were exactly searched for.

Example 1 [22]

Consider a function f:Z2n↦{0,1}⊂Z2, such that f(x)=1 if the sign of an element xX1 is to change by opposite, and f(x)=0 otherwise, where by definition Z2n:=X0∪X1,X0∩X1=∅. Take a test element |(a)>∈H(1) as follows: |(a)>:=12(|0>−|1>)and compute the value Ufx∈Z2nαx|(x)>⊗12(|0>−|1>)=12x∈Z2nαx|(x),0>−∑x∈Z2nαx|(x),1>=12x∈X0αx|(x),0>−∑x∈X0αx|(x),|1>+12x∈X1αx|(x),1>−∑x∈X1αx|(x),|0>=x∈X0αx|(x)>−∑x∈X1αx|(x)>12(|0>−|1>).It is seen well now that the Uf-operator (13) really changed all the signs at X1x-elements by the opposite.

Algorithms like that above can effectively solve as is well known [22] the following important for applications problems not solvable generally in a reasonable time by usual classical computers:

  • (i)

    factorization of a large integer x∈Z+ by its primes (Shor, 1994) and application of it to encrypting messages encoded via the RSA system;

  • (ii)

    search or sorting algorithm for finding an item in structured and unstructured data sets (Grover, 1996; Hogg, 1997);

  • (iii)

    fast discrete Fourier transform (Shor, 1994);

  • (iv)

    finding minimal periods of periodic functions (Shor, 1994; Kitayev, 1995) and other ones.

The important ingredient of quantum computing algorithms is the construction of corresponding unitary transformations (9) and further controlling its action on information data vector from the proper quantum computation medium.

Below we shall discuss some of the well known examples which were recently treated by means of quantum computing algorithms.

Example 2 Public key cryptography: (RSA)-cryptosystem

Take any message P which we want to send—being called a plaintext. Using a key to encipher, it into ciphertext C one can transmit it to the receiver who uses another key to decipher the C into P. A simple RSA scheme acts as follows: take any two large enough primes p and qinZ+ and let m:=pq. Knowing these p and q one can compute easily the Euler function ϕ(m)=(p−1)(q−1). Choose now any integer einZ+ coprime to ϕ(m), i.e. (ϕ(m),(e)=1 (the common divisor equals 1). Numbers m and e are published!

Consider a plaintext P∈Z+ (presented as a large integer) being less than m→Z+ and coprime to m, i.e. (P,m)=1. (N.B. Almost all integers P<m are evidently coprime). The corresponding ciphertext will be the unique integer C:=Pe(modm), 0<C<m. It is important: prime factors p,q and ϕ(m) are hidden from public (!). In order to decrypt the plaintext P from C, we need to get a receiver key—the integer d∈Z+, such that ed≡1(modϕ(m)), which can be found by using the Euclidean algorithm. Then, one has: Cd≡Ped≡P(1+kϕ(m))≡P(Pϕ(m))k≡P(1)(modm):→P,that is the plaintext P is decrypted!

Now we will demonstrate the quantum Shor’s counterpart of the algorithm described above. It is well known fact that the main problem being unsolvable by classical computer in a reasonable time is factoring an integer m∈Z+ by its primes. It is a classical result that the factorization problem reduces to finding the mth order r∈Z+ of an element a∈Z+ in Zm, i.e. ar≡1(modm),(a,m)=1. In general, m∈Z+ is taken to be odd as multiples of 2 are evidently not important, as well as mpe for some p∈Z+ and power e∈Z+. By definition, ae+sr≡ae(modm) for any integers e∈Z+,s∈Z, i.e. the function f(k):=ak(modm) is r-periodic! This period r∈Z+ can be found easily making use of the Shor’s QFT-algorithm, i.e. via the quantum Fourier transform. To make an introduction into this transform consider an abstract abelian group G={g1,g2,gn,…} like (Z,+),(Z,·),(Zn+,·) and so on, and denote by i:i=1,dimG} its characters, that is G-invariant functions on G in C. All G-invariant functions f:G→C form the linear space FG, on which there exists the standard G-invariant scalar product (f,h):=∑i=1dimGf(gi)h(gi). The corresponding norm of fFG is denoted as ||f||. The following lemma holds.

Lemma 1

Characters χi∈FG,i=1,dimG, form an orthogonal basis of FG, that is spanCi:i=1,dimG}=FG and ij)=δij,i,j=1,dimG. Then for any fFG one can write down the expansion f=∑i=1dimGciχi with some ciC,i=1,dimG.

Definition 2

A function f̂∈FG is called the discrete Fourier transform of fFG if ci=f̂(gi),i=1,dimG. From the above one follows that f̂(gi)=∑k=1dimGχi(gk)f(gk)foranyi=1,dimG.

As a result, the Parseval identity ||f||=||f|| holds for any fFG. Turn now back to our quantum Hilbert space H of a sufficient dimension dimH, such that all elements giG, i=1,dimG, can be represented by means of basis vectors |gi>H, i=1,dimG. A general element gG, ||g||=1, is evidently represented in H as |g>=∑i=1dimGci(g)|gi>, <gi|gj>= δij, i,j=1,dimG, where ∑k=1dimG|ci(g)|=1. One can now define a function f̂∈FG, such that f̂(gi)=ci(g), i=1,dimG, ||f̂||=1.

Definition 3

The quantum Fourier transform (QFT) is the operation i=1dimGf(gi)|gi>→∑i=1dimGf̂(gi)|gi>, where evidently f̂(gi)=∑k=1dimGχi(gk)f(gk) for any i=1,dimG.

As a simple consequence, the definition above states that the QFT is a linear and unitary transformation. One finds easily now that |gi>→∑k=1dimGχk(gi)|gk>, i=1,dimG, that is the corresponding unitary transformation matrix looks as Uχ:={χi(gj):i,j=1,dimG}.

Take now for instance G=Zn; dimG=n; the corresponding characters χy(x)=exp((2πi/n)xy), where x,y∈Zn. Then the corresponding QFT on Zn is the operation |x>QFT(1/n)∑y=0n−1exp(−(2πi)/(n)xy)|y> for any x,yinZn={0,1,2,…,n−1}. Proceed now to embedding Zn into H as a basis {|k>inH:k=0,n−1}. For this assume that n=n1n2 and (n1,n2)=1. Then by means of the classical Chinese remainder theorem the following decomposition holds: ZnZn1×Zn2 and the isomorphic mapping Φ:Zn1×Zn2Zn acts as follows: (k1,k2)=a1n2k1+a2n1k2, where a1 (resp. a2) is the multiplicative inverse of n2 in Zn1 (of n1 in Zn2). Having assumed now that QFT is available for Zn1 and Zn2, one can easily construct QFT for Zn!, taking into account that Un1:|k1>→(1/n1)∑l1=0n1−1exp(−(2πi/n1)l1k1)|l1>,Un2:|k2>→(1/n2)∑l2=0n2−1exp(−(2πi/n2)l2k2)|l2>, where Zn1×Zn2∋(k1,k2)⇆k=a1n2k1+a2n1k2Zn. Thus, one gets |k1,k2>→|a1k1,a2k2>=(Un1|k1>)⊗(Un2|k2>)=(1/n1n2)∑l1=0n1−1l2=0n2−1exp[−2πi/n1n2Φ(k1,k2)Φ(l1,l2]|Φ(l1,l2)>. This decomposition thereby can be applied further recursively both to n1 and to n2Z+.

Below we shall apply this construction for describing the Shor algorithm for finding the orders of integers. It consists of such steps.

  • Step 1. To prepare the superposition (1/m)∑k=0m−1|k,0>,n,m=2lZ+.

  • Step 2. To compute k→ak(modn) and to get due to the r-periodocity (1/m)∑k=0m−1|k,ak>=(1/m)∑l=0r−1q=0sl|qr+l,al>, where slmax{s:sr+l≤m},(m/r)−1−(l/r)≤sl≤(m/r)−(l/r).

  • Step 3. Compute the inverse QFT on Zm to get (1/m)∑l=0r−1q=0sl(1/m)∑p=0m−1exp[(2πpi/m)(qr+l)]|p,al>=(1/m)∑l=0r−1p=0m−1e(2πip/m)lq=0sle(2πip/m)rq|p,al>.

  • Step 4. To get some p∈Zn from the above mentioned expression.

  • Step 5. Find the continuous fraction expansion of p/m in the form (pi/qi)→(p/m),i∈Z+, making use of the Euclidean algorithm, and at last output the smallest qı̂Z+ such that aqı̂≡1(modn) if such one exists.

Take now, for example, integer n=15 and a=7 be the element whose period is to be found. Choose also m=16=24>15, and evidently a=7∈Z15. The group G=Z16, its dimension dimG=16 and the Hilbert space H=H(4). Then we get step by step:
  • Step 1. (1/4)∑k=015|k,0>∈H.

  • Step 2. k→7k(mod15) gives (1/4)(|0,1>+|4,1>+|8,1>+|12,1>+|1,7>+|5,7>+|9,7>+|13,7>+|2,4>+|6,4>+|10,4>+|14,4>+|3,13>+|7,13>+|11,13>+|15,13>).

  • Step 3. The inverse QFT gives in Z16:(1/4)(|0,1>+|4,1>+|8,1>+|12,1>+|0,7>+i|4,7>−|8,7>−i|12,7>+|0,4>−|4,4>+|8,7>−|12.4>+|0,13>−i|4,13>−|8,13>+i|12,13>).

  • Step 4. Choose p-elements p1=0,p2=4,p3=8,p4=12, which are present in the above expansion.

  • Step 5. The corresponding convergents for (4/16)=(1/4) are {(0/1),(1/4)}, for (12/16)=(3/4) are {(0/1),(1/1),(3/4)}, giving rise to the correct period r=4; thereby one has 7r−1=74−1≡0(mod15) and 7(r/2)−1=7(4/2)−1≡3(mod15),7(4/2)+1≡5(mod15), or 3=gcd(3,15),5=gcd(5,15) are the true nontrivial factors of 15.

Section snippets

Loop Grassmann manifolds

The following observation due to [11], [26] appeared to be fruitful for devising a new approach for realization of quantum computing algorithms. Namely, take any self-adjoint projector operator P:HH,P2=P, and construct the operator U:=I−2P.Then evidently, U+U=I, that is the mapping U:HH is unitary. It is called [11] a uniton being of great importance for constructing so called quantum computing gates [22]. Since these projection operators, being in general dependent on some parameter space,

Symplectic structures on loop Grassmann manifolds and Casimir invariants

A point P∈CS1(M) satisfying by definition the quadratic constraint P2=P due to Lemma 3, can be naturally imbedded into the adjoint space CS1(u(H)), thereby defining the following principal fiber bundle: where we have denoted by G:=CS1(U(H)),g-the corresponding adjoint space to the Lie algebra g:=CS1(u(H)) with respect to the ad-invariant symmetric and nondegenerate Killing form [1], π:g→CS1(M) -the corresponding natural projection onto CS1(M) generated by the constraint l2+il=0 for l∈g due

An intrinsic loop Grassmannian structure and dual momentum mappings

As is well known [10], [21], [17] a large class of integrable dynamical systems on an infinite dimensional manifold can be successfully derived by constructing a momentum mapping into the dual spaces of some loop Lie algebras via the Lie-algebraic approach. To proceed with such a construction in the case of our loop Grassmann manifold CS1(M), let us represent the manifold CS1(M) as a manifold reduced from the matrix manifold C̃S1(M):=CS1(Mm,N)×CS1(Mm,N), endowed with a canonical symplectic

Holonomy group structure of the quantum computing medium

Consider the fiber bundle G(m)×C̃S1(M)π̃CS1(M), where by definition π̃(F,F̃):=F+F∈CS1(M)for any (F,F̃)∈C̃S1(M). Since the manifold C̃S1(M) is symplectic with the symplectic structure (36), one can reduce it upon the loop Stiefel manifold StS1(M) [25] subject to the momentum mapping q:C̃S1(M)→g(m) having put q(F,F)=0,Sp(FF+)=m. Thereby, we get that StS1(M)={(F,F+)∈C̃S1(M):FF+=Im}.Since the loop group G(m) acts also on the Stiefel manifold StS1(M) leaving it invariant, we can

Two-mode quantum–optical model

As is shown in [15], [11] there exists a possibility to realize so called optical holonomic quantum calculations based on the structure group G(m)=CS1(U(m)),m∈Z+. For this case, at m=4, let us consider a self-adjoint operator Ĥ0:→H(2)H(2), dimCH(2)=4, in the standard quantum optical form Ĥ0=N̂1(N̂1−1)⊕N̂2(N̂2−1)with factors N̂1:=a1+a1,N̂2:=a2+a2,realized by means of the usual [19] creation and annihilation bose-operators aj,ak+:H(2)H(2),j,k=1,2, satisfying the standard commutator

Acknowledgements

One of the authors (A.P.) is grateful to Prof. Anatoliy Nikitin (IM NAN, Kiev, Ukraine) for his kind invitation to report on the differential geometric and informatics aspects of modern quantum computation theory, at the International Conference “Symmetry-2003” which was held in Kiev, Ukraine. The author is also grateful to Prof. J. Klamka for reading the manuscript and suggesting some important propositions which have substantially improved the article.

(U.T.) is grateful to UW-Madison USA for

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