aa r X i v : . [ qu a n t - ph ] J a n On swapping the states of two qudits
Colin M Wilmott
Institut f¨ur Theortische Physik III, Heinrich-Heine-Universit¨at, D¨usseldorf,GermanyE-mail: [email protected]
Abstract.
The SWAP gate has become an integral feature of quantum circuitarchitectures and is designed to permute the states of two qubits through the useof the well-known controlled-NOT gate. We consider the question of whether atwo-qudit quantum circuit composed entirely from instances of the generalisedcontrolled-NOT gate can be constructed to permute the states of two qudits.Arguing via the signature of a permutation, we demonstrate the impossibility ofsuch circuits for dimensions d ≡
1. Introduction
A prerequisite for quantum computation is the successful implementation of multiple-qubit quantum gates. The most elementary of all multiple-qubit quantum gates ispremised by two-qubit controlled unitary operators, and a classic example is thecontrolled-
NOT ( CNOT ) gate. The
CNOT gate has been shown to provide a basisfor a measurement set that permits the construction of syndrome tables used in errorcorrection (Chuang & Yamamoto 1997). Additionally, the
CNOT gate has assumeda central role in the theory of quantum computation. It is the quantum mechanicalanalogue of the classical connective
XOR gate and is a principle component in universalcomputation. Furthermore, Barenco et al. (1995) have shown that any multiple-qubit quantum operation may be restricted to compositions of single-qubit gates andinstances of the
CNOT gate. It is for this reason that we say the quantum gate libraryconsisting of single-qubit gates and the
CNOT gate is universal. As a consequence,the
CNOT gate has acquired the special status as the hallmark of multi-qubit control(Vidal & Dawson 2004).In recent years, researchers in universal quantum computation have doneconsiderable work optimizing quantum circuit constructions. Vatan & Williams(2004) constructed a quantum circuit for general two-qubit operations which requiresfifteen single-qubit gates and at most three
CNOT gates. A crucial aspect of thisresult is the demand that the
SWAP gate requires at least three
CNOT gates. The
SWAP gate describes the cyclical permutation of the states of two qubits and hasbecome an integral feature of the circuitry design for many quantum operators. Itis a fundamental element in the circuit implementation of Shor’s algorithm (Fowleret al. 2004), and Liang & Li (2005) maintain that experimentally realising the
SWAP gate is a necessary condition for the networkability of quantum computation. n swapping the states of two qudits ht | ψ i| φ i | ψ i| φ ⊕ ψ i (b) th | ψ i| φ i | ψ ⊕ φ i| φ i Figure 1.
Circuit descriptions for the CNOT gate types. (a) The CNOT1 gate;the state of control system | ψ i ∈ H A remains unchanged after application whereasthe state of the target system | φ i ∈ H B is transformed under modular arithmeticto the state | φ ⊕ ψ i . (b) The CNOT2 gate in which the roles of systems H A and H B are reversed. ht th ht | ψ i| φ i | φ i| ψ i Figure 2.
The
SWAP gate illustrating the cyclical permutation of two qubits.System A begins in the state | ψ i and ends in the state | φ i while system B beginsin the state | φ i and ends in the state | ψ i . In this paper, we examine the possibility of constructing a two-qudit
SWAP gateusing only instances of the generalised
CNOT gate to permute the states of two qudits.Section 2 introduces preliminary material from the theory of permutations whichwill serve as a basis for our study. Section 3 considers the particular problem ofwhether a two-qutrit quantum circuit composed entirely from instances of the two-qutrit
CNOT gate can be constructed to permute the states of two qutrits. Finally,section 4 generalises the results of section 3 to two-qudit quantum systems beforedemonstrating the impossibility of two-qudit
SWAP gates using only instances of thegeneralised
CNOT gate for dimensions d ≡
2. Preliminaries
Let H denote the d -dimensional complex Hilbert space C d . Fix each orthonormalbasis state of the d -dimensional Hilbert space to correspond to an element of Z d ; assuch the basis {| i , | i , . . . , | d − i} ⊂ C d whose elements correspond to the columnvectors of the identity matrix I d is called the computational basis. A qudit is a d -dimensional quantum state | ψ i ∈ H written as | ψ i = P d − i =0 α i | i i where α i ∈ C and P d − i =0 | α i | = 1. Given d -dimensional Hilbert spaces H A and H B , consider the set of d × d unitary transformations U ∈ U ( d ) that act on the two-qudit quantum system H A ⊗ H B . Let U CNOT1 ∈ U ( d ) represent the generalised CNOT gate that has controlqudit | ψ i ∈ H A and target qudit | φ i ∈ H B . The action of U CNOT1 on the set of basis n swapping the states of two qudits | m i ⊗ | n i of H A ⊗ H B is given by U CNOT1 | m i ⊗ | n i = | m i ⊗ | n ⊕ m i , m, n ∈ Z d , (1)with ⊕ denoting addition modulo d . Similarly, let U CNOT2 ∈ U ( d ) denote thegeneralised CNOT gate having control qudit | φ i ∈ H B and target qudit | ψ i ∈ H A .The action of U CNOT2 on the set of basis states | m i ⊗ | n i of H A ⊗ H B is written U CNOT2 | m i ⊗ | n i = | m ⊕ n i ⊗ | n i , m, n ∈ Z d . (2)Figure 1 provides the quantum gate circuitry representation for the respective CNOT types while figure 2 illustrates the well-known
SWAP gate for permuting the states oftwo qubits.
Consider the set N = { , , . . . , n } and let σ : N N be a bijection. Let σ = (cid:20) . . . ni i . . . i n (cid:21) be a permutation of the set N with i k ∈ N denoting theimage of k ∈ N under σ . Let σ and τ be two permutations of N. Define the product σ · τ by ( σ · τ )( i ) = σ ( τ ( i )), i ∈ N, to be the composition of the mapping τ followed by σ . These permutations taken with ( · ) form the group S n called the symmetric group of degree n .Given the permutation σ ∈ S n and for each i ∈ N, let us consider the sequence i, σ ( i ) , σ ( i ) , . . . . Since σ is a bijection and N is finite there exist a smallest positiveinteger ℓ = ℓ ( i ) depending on i such that σ ℓ ( i ) = i. The orbit of i under σ thenconsists of the elements i, σ ( i ) , . . . , σ ℓ − ( i ). By a cycle of σ , we mean the ordered set( i, σ ( i ) , . . . , σ ℓ − ( i )) which sends i into σ ( i ), σ ( i ) into σ ( i ), . . . , σ ℓ − ( i ) into σ ℓ − ( i ),and σ ℓ − ( i ) into i and leaves all other elements of N fixed. Such a cycle is called an ℓ -cycle. We refer to 2-cycles as transpositions and note that any permutation can bewritten as a product of transpositions. A pair of elements { σ ( i ) , σ ( j ) } is an inversion in a permutation σ if i < j and σ ( i ) > σ ( j ). The number of transpositions in anysuch product is even if and only if the number of inversions is even. Consequently, wesay such a permutation is even. A similar case holds for odd permutations. Lemma 2.1
Every permutation can be uniquely expressed as a product of disjointcycles.
Proof.
Let σ be a permutation. Then the cycles of the permutation are of the form i, σ ( i ) , . . . , σ ℓ − ( i ). Since the cycles are disjoint and by the multiplication of cycles, wehave it that the image of i ∈ N under σ is the same as the image under the product, ς , of all the disjoint cycles of σ . Then σ and ς have the same effect on every elementin N , hence, σ = ς . (cid:3) Every permutation σ ∈ S n has a cycle decomposition that is unique up to theordering of the cycles and up to a cyclic permutation of the elements within eachcycle. Further, if σ ∈ S n and σ is written as the product of disjoint cycles of length n , . . . , n k , with n i ≤ n i +1 , we say ( n , . . . , n k ) is the cycle type of σ . As a result ofLemma 2.1, every permutation can be written as a product of transpositions. Sincethe number of transpositions needed to represent a given permutation is either evenor odd, we define the signature of a permutation assgn( σ ) = (cid:26) +1 if σ is even − σ is odd. (3) n swapping the states of two qudits σ ∈ S n , let us consider the corresponding permutation matrix A σ whereby A σ ( j, i ) = (cid:26) σ ( i ) = j f : S n det( A σ ) wheredet( A σ ) = X σ ∈ S n sgn( σ ) n Y i =1 A σ ( i ) ,i (5)is a group homomorphism. The kernel of this homomorphism, ker f , is the set of evenpermutations. Consequently, we have it that σ ∈ S n is even if and only if det( A σ )equals +1.
3. On swapping the states of two qutrits
Let d = 3 and consider the following problem. Given a pair of qutrit quantum systems,system A in the state | ψ i and system B in the state | φ i , and using only instancesof the two-qutrit CNOT gate, determine if it is possible permute the states of thecorresponding systems so that system A ends in the state | φ i while system B ends inthe state | ψ i . Problem 3.1
Given qutrits | ψ i ∈ H A and | φ i ∈ H B and using only instancesof the two-qutrit CNOT gate, determine if it is possible to construct a two-qutritquantum circuit that permutes the states of the quantum systems H A and H B suchthat | ψ i ⊗ | φ i ∈ H A ⊗ H B is mapped to | φ i ⊗ | ψ i ∈ H A ⊗ H B . We now show that for a pair of qutrits, it is not possible to permute the states | ψ i ⊗ | φ i ∈ H A ⊗ H B using only instances of the two-qutrit CNOT gate.Any two-qutrit quantum circuit composed entirely from instances of the two-qutrit
CNOT gate can be written in terms of the two-qutrit
CNOT1 and
CNOT2 gates. The action of the two-qutrit
CNOT1 gate on the basis states | m i ⊗ | n i ∈H A ⊗ H B , m, n ∈ Z , is described by the unitary transformation U CNOT1 ∈ U (9) givenby U CNOT1 | m i ⊗ | n i = | m i ⊗ | n ⊕ m i , m, n ∈ Z , (6)where ⊕ denotes addition modulo 3. Figure 3 (a) provides the matrix descriptionfor the two-qutrit CNOT1 gate. A similar description for the two-qutrit
CNOT2 gateholds, and figure 3 (b) provides the corresponding matrix description. Note also thatthe two-qutrit
CNOT1 and
CNOT2 gates can be described in the following way. Thepermutation matrix corresponding to the two-qutrit
CNOT1 gate takes the value 1 inrow 3 m + n and column 3 m + ( n ⊖ m ) , m, n ∈ Z . Similarly, the matrix correspondingto the two-qutrit
CNOT2 gate takes the value 1 in row 3 m + n column 3( m ⊖ n ) + n , m, n ∈ Z . Importantly, both of these matrix descriptions have determinant +1 asthe permutations corresponding to their the respective matrices are even.Let us now assume there exists a two-qutrit quantum circuit composed entirelyfrom instances of the two-qutrit CNOT gate types which permutes the states of twoqutrits. By assumption, such a circuit will then be a composition of two-qutrit
CNOT1 and
CNOT2 gates. It then follows that any composition of two-qutrit
CNOT1 and
CNOT2 gates will be equivalent to some product of their respective unitary matrixdescriptions. Such a matrix product will necessarily have determinant +1 as both n swapping the states of two qudits (b)
Figure 3.
Matrix representations of two-qutrit CNOT types. (a) The matrixrepresentation for the two-qutrit CNOT1 gate. (b) The matrix description forthe two-qutrit CNOT2 gate.
Figure 4.
The matrix U ∗ ∈ U (9) that permutes the states of two qutrits. constituent elements have determinant +1. However, figure 4 represents the unitarytransformation U ∗ ∈ U (9) required to permute the states of two qutrits. Such aswap matrix takes the value 1 in row 3 m + n column 3 n + m , and has determinant −
1. Therefore, no composition of two-qutrit
CNOT gate types can yield the requiredmatrix, and the result follows. (cid:3) n swapping the states of two qudits
4. On swapping the states of two qudits
We generalize problem 3.1 to higher dimensional quantum systems and ask if it ispossible to construct a two-qudit quantum circuit composed entirely from instancesof the generalised
CNOT gate to permute the states of two qudit quantum systems.
Problem 4.1
Given a pair of qudits | ψ i ∈ H A and | φ i ∈ H B and using only instancesof the generalised CNOT gate, determine if it is possible to construct a two-quditquantum circuit to permute the state | ψ i ⊗ | φ i ∈ H A ⊗ H B . We have shown in section 3 that the unitary matrices corresponding to the two-qutrit
CNOT1 and
CNOT2 gate types both have determinant +1, and this contrastedsignificantly with the determinant of the unitary matrix required to permute the statesof a pair of qutrits. Consequently, no composition of the former could yield thelatter and the result followed. There is, however, another way to look the problem ofpermuting the states of two quantum systems using only instances of the generalised
CNOT gate, and it is the following. Firstly, in examining the qutrit case, we notethat the permutations corresponding to the two-qutrit CNOT1 matrix and the swapmatrix U ∗ ∈ U (9) are σ CNOT1 = (cid:20) (cid:21) (7) σ U ∗ = (cid:20) (cid:21) (8)respectively. In particular, these permutations describe, respectively, the action ofboth the two-qutrit CNOT1 gate and the swap matrix U ∗ on the set of basis states | m i ⊗ | n i ∈ H A ⊗ H B , m, n ∈ Z . The cycle type for two-qutrit CNOT gate is(1 , , , ,
3) while the cycle type for the swap matrix U ∗ is (1 , , , , , CNOT gate fixes three basis states and permutes the remaining states in twocycles of length 3. Each such cycle may be written as a product of two transpositions.Whence, the signature of the two-qutrit
CNOT permutation is +1. On the other hand,the permutation describing swap of a pair of qutrit states contains three fixed elementsand a set of three transpositions and therefore the signature of this permutation is −
1. Consequently, it follows that within a two-qutrit quantum circuit architecture, nocomposition of the two-qutrit
CNOT gate types alone can permute the states of twoqutrits.More generally, the two-qudit
CNOT gate that acts on a pair of d -dimensionalquantum systems corresponds to a permutation of the d basis states. For primedimensions d = p , the permutation associated with the generalised CNOT1 gate fixes d basis states and induces ( d −
1) cycles of length d , each of which may be writtenas a product of ( d −
1) transpositions. The generalised
CNOT1 gate permutation isthen a composition of ( d − basis state transpositions. A similar case holds for thegeneralised CNOT2 gate in that corresponding mapping fixes d basis elements induces( d −
1) cycles where each is a product of ( d −
1) transpositions. Therefore, the signatureof the generalised
CNOT permutation is − d = 2 and +1 for odd primedimensions.Now, let us consider the unitary matrix U ∗ ∈ U ( d ) that swaps the states of twoqudits. This matrix permutes the basis states of a pair of qudits thereby mappingthe two-qudit state | ψ i ⊗ | φ i ∈ H A ⊗ H B to the state | φ i ⊗ | ψ i ∈ H A ⊗ H B . Sucha transformation corresponds to a permutation of the d basis states | m i ⊗ | n i ∈ n swapping the states of two qudits H A ⊗ H B , m, n ∈ Z d . Under this mapping, there are d fixed basis elements and d ( d − / SWAP gate is − d ≡ d ≡ CNOT gate cannot permute the states of two qudits when d ≡ (cid:3)
5. Conclusion
We considered the problem of constructing a two-qudit
SWAP gate using only instancesof the generalized
CNOT gate. We discussed the idea of signature for a permutationand identified, via this signature, when it is possible to permute the states of twoqudits using only instances of the generalised
CNOT gate. Based on this argument,we demonstrated the impossibility of constructing a two-qudit
SWAP gate using onlyinstances of the generalised
CNOT gate for dimensions d ≡ k -qudit SWAP gate usingonly generalised CNOTs . Finally, our understanding of the symmetric group and itsdecompositions may well be enhanced by considering how to design quantum circuitsto realise certain subgroups of this group.
Acknowledgments
The author wishes to thank Prof. Peter Wild for many helpful suggestions.
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