A geometrical approach to quantum holonomic computing algorithms
Introduction
Consider a two-dimensional vector subspace of some complex Hilbert space called further a quantum bit (qubit) information space as follows: with orthonormal base vectors |0>,|1> subject to the scalar product 〈·,·〉 in and also its nth tensor product being a Hilbert space of dim , 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 : that is we will calculate all values , at the same time (with needed accuracy).
Take any and represent it in two-digital {0,1}-system, thereby we can write down the bijection where by definition for any . Such a sequence (x1,x2,x3,…,xn)2 can now be bijective imbedded into the above constructed Hilbert space of n-qubit states: So, for any one can construct an n-qubit state via the induced mapping On the set one has the usual algebraic operations: if , then or Definition 1 The following unitary transform of the Hilbert space for a mapping , 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 Example 1 [22] Consider a function , such that f(x)=1 if the sign of an element x∈X1 is to change by opposite, and f(x)=0 otherwise, where by definition . Take a test element as follows: and compute the value It is seen well now that the Uf-operator (13) really changed all the signs at X1∋x-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: factorization of a large integer by its primes (Shor, 1994) and application of it to encrypting messages encoded via the RSA system; search or sorting algorithm for finding an item in structured and unstructured data sets (Grover, 1996; Hogg, 1997); fast discrete Fourier transform (Shor, 1994); 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 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 coprime to ϕ(m), i.e. (ϕ(m),(e)=1 (the common divisor equals 1). Numbers m and e are published!
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 by its primes. It is a classical result that the factorization problem reduces to finding the mth order of an element in , i.e. . In general, is taken to be odd as multiples of 2 are evidently not important, as well as m≠pe for some and power . By definition, for any integers , i.e. the function is r-periodic! This period 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 like and so on, and denote by its characters, that is G-invariant functions on G in . All G-invariant functions form the linear space FG, on which there exists the standard G-invariant scalar product . The corresponding norm of f∈FG is denoted as ||f||. The following lemma holds.
Lemma 1
Characters , form an orthogonal basis of FG, that is and . Then for any f∈FG one can write down the expansion f=∑i=1dimGciχi with some .
Definition 2
A function is called the discrete Fourier transform of f∈FG if . From the above one follows that
Definition 3
The quantum Fourier transform (QFT) is the operation , where evidently for any .
Take now for instance ; dimG=n; the corresponding characters χy(x)=exp((2πi/n)xy), where . Then the corresponding QFT on is the operation for any . Proceed now to embedding into as a basis . For this assume that n=n1n2 and (n1,n2)=1. Then by means of the classical Chinese remainder theorem the following decomposition holds: and the isomorphic mapping acts as follows: (k1,k2)=a1n2k1+a2n1k2, where a1 (resp. a2) is the multiplicative inverse of n2 in (of n1 in ). Having assumed now that QFT is available for and , one can easily construct QFT for , taking into account that , where . Thus, one gets . This decomposition thereby can be applied further recursively both to n1 and to .
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 .
Step 2. To compute and to get due to the r-periodocity , where .
Step 3. Compute the inverse QFT on to get .
Step 4. To get some from the above mentioned expression.
Step 5. Find the continuous fraction expansion of p/m in the form , making use of the Euclidean algorithm, and at last output the smallest such that if such one exists.
Step 1. .
Step 2. 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 .
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 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 , and construct the operator Then evidently, , that is the mapping 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 satisfying by definition the quadratic constraint P2=P due to Lemma 3, can be naturally imbedded into the adjoint space , thereby defining the following principal fiber bundle: where we have denoted by -the corresponding adjoint space to the Lie algebra with respect to the ad-invariant symmetric and nondegenerate Killing form [1], -the corresponding natural projection onto generated by the constraint l2+il=0 for 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 , let us represent the manifold as a manifold reduced from the matrix manifold , endowed with a canonical symplectic
Holonomy group structure of the quantum computing medium
Consider the fiber bundle , where by definition for any . Since the manifold is symplectic with the symplectic structure (36), one can reduce it upon the loop Stiefel manifold [25] subject to the momentum mapping having put . Thereby, we get that Since the loop group G(m) acts also on the Stiefel manifold 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 . For this case, at m=4, let us consider a self-adjoint operator , dim, in the standard quantum optical form with factors realized by means of the usual [19] creation and annihilation bose-operators , 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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