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In this paper, I study the invariant subspaces of quantum states under
SWAP^{α} gates arising from the exchange interaction and their use in quantum computation. I investigate the generation and characterization of invariant-subspace vector-states that arise from such gates. I also state a condition for the locus of states that are accessible using the
SWAP^{α} gates, given an initial input state.

The non-local physical phenomena related to quantum entanglement [

A form of symmetry that has been of significant interest to the world of quantum information is that of symmetry under permutations [^{α} gate, together with single-qubit operations, is found to produce a universal set of quantum gates (enabling any quantum computation) [^{α} gate [

Burkard et al. [^{α} gates if one encodes pseudo-spin qubits using three physical qubits [^{α} with a specific value of α. It was shown by Tanamoto et al. [

preparing the initial state as | ± 〉 = 1 2 ( | 0 〉 + | 1 〉 ) ( | 0 〉 − | 1 〉 ) and applying the SWAP one can obtain Raussendorf’s cluster state [

In this paper, I am interested in the general question of what kinds of states are accessible from an arbitrary quantum state, which is easier to produce in physical systems than entangled states, using the SWAP^{α} operators and what are their applications? Our approach to this is to determine the invariant subspaces of the permutation group S n when applied to the Hilbert space of n-qubits. These subspaces comprise vectors that map back onto a linear superposition of the same vectors under the action of SWAP^{α} operators. In Section II, we look more closely at the invariant subspaces for the combinations of SWAP^{α} gates and look at the kinds of invariant subspaces that can arise for quantum states with different numbers of | 0 〉 and | 1 〉 in them. In Section III, we highlight the numerical complexity of the problem and look closely at the analytical method to generate vector states for our quantum system. In Section IV, we look at the idea of symmetry under parity of this system and discuss an efficient algorithm for generating n-qubit vector states for SWAP^{α} gates. In Section V, we look at the problem of accessibility of quantum states using Power-of-SWAP gates and devise conditions for the output states, given aninput state and combination of SWAP^{α} gates.

Quantum logic gates are essential building blocks of a quantum computer. The SWAP^{α} quantum gate with 0 < α ≤ 1 is one of the most efficient quantum gates in two-qubit quantum computation, with three SWAP^{α} gates combined with six single-qubit gates being able to realize any arbitrary two-qubit unitary operation [^{α} gate can be experimentally implemented in several physical systems such as in the exchange interaction between electrons trapped by surface acoustic waves [^{α} operators for generation of quantum states and our analysis applies to any n-qubit quantum state that can undergo SWAP^{α} operations, where α is any real number.

We find that a combination of all SWAP^{α} gates between a finite number of qubits comprise a group that is isomorphic to the Symmetric Group , which is the group of all permutations or self-bijections of a set of elements with the operation of composition, and that only certain points in the Hilbert Space are accessible using the SWAP^{α} gates, given a particular input state. To find the kinds of states that are accessible under the repeated action of a SWAP^{α} operator, we need to study the invariant subspaces^{1} associated with this combination of SWAP^{α} operators. We find the structure of the invariant subspaces of the combination of SWAP^{α} gates to be the same for all α.

The invariant subspaces of a vector space under the action of a group emerge from the irreducible representations of the group. For any group, the number of irreducible representations is equal to the number of conjugacy classes^{2}. It is seen that for the Symmetric Group S n , the way in which we can permute n elements without changing the structure of the superposition of vector states characterize the specific conjugacy class, and by extension the invariant subspace of the group, associated with those vector states. In other words, the invariant subspaces of the Symmetric Group S n comprise of vector states, whose linear superposition remains invariant under the action of permutation of elements within the partition that corresponds to the invariant subspace. For instance, in the case of permutation symmetries of four elements, we have conjugacy classes of the Symmetric Group S 4 such as [^{3}. A point to note here is that there could be several cases for each cycle pattern. For instance, for four qubits 1-2-3-4, (12) (34) and (13) (24) are specific [^{th} and j^{th} qubit.

All the vector states that are associated with any specific invariant subspace corresponding to the Symmetric Group S n are found to have the same Hamming weight in their qubit representation. We find that the number of basis vectors comprising a specific invariant subspace and the forms of these vector states are related to the kind of permutation permissible within the partition corresponding to the invariant subspace. For n-qubit states, those vector states that span the entire subspace of states with a particular Hamming weight in as close to an equiprobabilistic manner (over the basis vectors) as possible are found to be closer to being invariant under all possible permutations of the n qubits, and are found to be part of one-dimensional invariant subspaces (with only one constituent vector state). For instance, the three-qubit quantum state that is invariant under all permutation under S 3 of its qubits is

| ψ 〉 = 1 3 ( | 001 〉 + | 010 〉 + | 100 〉 ) and this is the only vector comprising the

invariant subspace corresponding to the trivial irreducible representation for three-qubit states with Hamming weight 1 in their qubit representation.

The number of | 0 〉 and | 1 〉 qubits in multiqubit states, represented by their Hamming weight, is found to uniquely define the kinds of invariant subspaces, with their associated permutation symmetries and corresponding partitions, of the SWAP^{α} gates we can create with those states.

Let us consider a quantum state of the form

| ψ 〉 = ∑ PERM ( i ) α i | 00 ⋯ 0 ︸ k 11 ⋯ 1 ︸ n − k 〉 i (1)

where i is an index related to each permutation of the qubits and the summation is over i, all the permutations of the qubits. We find that only certain permutations (and ways of partitioning) are applicable and important for a given set of vector-states that have a specific value of k. The change in an arbitrary superposition of the vector states resulting from these specific permutation operations can be used to express the change in the superposition of these vector states under any other permutation operation. In this respect, for n-qubit states with even

n and k ≤ n 2 , we find that the relevant partitions (and corresponding permutation symmetries) are C ( k ) = { C i ( k ) } = { [ n ] , [ n − 1 , 1 ] , ⋯ , [ n − k , k ] } , where the i is

the index of the partitions for a fixed value of $k$. This is because the symmetries under permutations [ n − ( k + 1 ) , k + 1 ] , ⋯ , [ 11 ⋯ ( n times ) ] are encapsulated in the existing set C i . An example of this is shown in

partition [ n − 2 , 2 ] and so on till we have 1 set for the partition [ n 2 , n 2 ] , with there being n 2 + 1 kinds of subspaces that are relevant. As a result, N i n v ( even n ) = 1 2 ( n 2 + 1 ) ( ( n + 1 ) + 1 ) = 1 4 ( n + 2 ) 2 . For odd n, we have conjugacy classes C i ( k ) = { [ n ] , [ n − 1 , 1 ] , ⋯ , [ n − k , k ] } for k ≤ n − 1 2 and C i ( k ) = { [ n ] , [ n − 1 , 1 ] , ⋯ , [ n − ( n − k ) , n − k ] } for k ≥ n + 1 2 . We have ( n + 1 ) basis vector-sets associated with the invariant subspace [ n ] , ( n − 1 ) for [ n − 1 , 1 ] , ( n − 3 ) for [ n − 2 , 2 ] and so on till 2 sets for [ n + 1 2 , n − 1 2 ] , with there being n − 1 2 + 1 kinds of partitions that need be independently described and are relevant. As a result, N i n v ( odd n ) = 1 2 ( n − 1 2 + 1 ) ( ( n + 1 ) + 2 ) = 1 4 ( n + 1 ) ( n + 3 ) . Thus, we find that the number of invariant subspaces, N i n v , for an n-qubit system, operated upon by the SWAP^{α}, is given by:

N i n v = { 1 4 ( n + 2 ) 2 , if n is even 1 4 ( n + 1 ) ( n + 3 ) , if n is odd , n > 1 . (2)

Invariant subspaces of the Symmetric Group S n and reducibility of representations are closely linked [

U { SWAP } ( | q { i } 〉 , | q { j } 〉 ) as the SWAP operation between the i_{th} and j_{th} qubits in an n-qubit state | q 1 q 2 ⋯ q { i } ⋯ q { j } ⋯ q { n } 〉 , we find that the matrix representation U { SWAP } ( | q { i } 〉 , | q { j } 〉 ) can be transformed into a block form D, comprising of

irreducible representations of the Symmetric Group S n associated with each invariant subspace, with a similarity (equivalence) transformation S. If one has r distinct SWAP operations between qubits applied consecutively, then the resultant operation can be block-diagonalized as follows:

D = D 1 D 2 ⋯ D r = ( S − 1 U SWAP 1 S ) ( S − 1 U SWAP 2 S ) ⋯ ( S − 1 U SWAP r S ) (3)

where U SWAP i represent the i^{th} distinct SWAP operation.

This similarity transformation is found to be useful in block-diagonalizing the basic SWAP^{α}-gates representations as well, since by definition,

U SWAP α = 1 − e i π α 2 U SWAP + 1 + e i π α 2 I ( 2 × 2 ) (4)

where I ( 2 × 2 ) is a 2 × 2 identity matrix. As a result, the similarity transformation S that transforms the composite SWAP operator to a block-diagonal form: D = S − 1 U SWAP S , will also transform the U SWAP α in the same block form, up to

certain additive terms: S − 1 U SWAP α S = 1 − e i π α 2 D 1 + 1 + e i π α 2 I ( 2 × 2 ) ≡ D ′ . Due to the

relation established in (2), the similarity transformation and associated invariant subspace associated with U SWAP α must be of the same structural form as those for the U SWAP gate. This is also true for a combination of U SWAP α gates: ( S ⊗ n ) − 1 ( U SWAP α ⊗ n ) ( S ⊗ n ) = ( S − 1 U SWAP α S ) ⊗ n .

The matrix decomposition thus formed is block diagonal, as shown in the illustrative example in ^{α} gates, operating over a multi-qubit input state, produce a group that is isomorphic to the symmetric group S n . In fact, any function of SWAP^{α} gates can be put into the block form using

the same similarity transformation: S − 1 f ( U SWAP α ) S = f ( 0 ) S − 1 I 2 × 2 S +

f ′ ( 0 ) S − 1 U SWAP α S + f ″ ( 0 ) 1 2 ! S − 1 U SWAP α S S − 1 U S W A P α S + ⋯ , using Taylor series

expansion of the function of U SWAP α , and pre- and post-multiplying with S and S − 1 respectively. As a result, the determination of the similarity transformation for SWAP alone gives us a useful tool for finding the invariant subspaces and associated vectors for any arbitrary combination or function of partial SWAP gates.

Due to the non-commutativity of adjacent SWAP^{α} gates with at least one common qubit, the number of output states generated for a given input state and gate-combination is a lot more than in the case of commutative gates. For an l-qubit separable input state and k gates used, we have the following number of cases and associated accessible output states:

N a c c l = l k × 2 l . (5)

As can be seen, the problem grows exponentially with the number of qubits in the input (separable) states. The number of gates also scales this value exponentially. Certain cases are degenerate but the problem, all in all, grows exponentially with the number of qubits in the input state and the number of gates being operated with on the input state. As a result, this is a problem that cannot be treated efficiently using numerical methods, as the number of qubits involved increase. To resolve this problem, we look at analytic methods, using techniques for the representation theory for the symmetric group S n .

The system we are considering is an arbitrary n-qubit state and we are operating combinations of SWAP^{α} operations on such an input state. The action of

the is: | 00 〉 → | 00 〉 , | 01 〉 → 1 + e { i π α } 2 | 01 〉 + 1 − e { i π α } 2 | 10 〉 ,

| 10 〉 → 1 + e { i π α } 2 | 10 〉 + 1 − e { i π α } 2 | 01 〉 , | 11 〉 → | 11 〉 . The relevance of the SWAP or

partial SWAP arises from the fact that any permutation of elements can be expressed as a combination of transpositions [

The next significant step in the process of generating the vector states is to look at the irreducible representations { D i , i = 1 , 2 , ⋯ , N ! } of the group. The first step in this regard is to find the dimensions of the irreps using the Young’s Tableau [

d = n ! h . (6)

Once we have found the dimensions of the irreps, the next step is to find the forms of the irreps. We find the Young’s orthogonal form for the irreps using the equation [

D [ μ ] ( ( k , k + 1 ) ) { e } M 1 ⋯ M n [ μ ] = ( ρ ( M 1 ⋯ M n ; k + 1 , k ) ) { − 1 } e ^ M 1 ⋯ M n [ μ ] + A e ^ M 1 ⋯ M k + 1 M k ⋯ M n [ μ ] , (7)

where A = 1 − ( ρ ( M 1 ⋯ M n ; k + 1 , k ) ) − 2 and where we use the orthonormal basis e ^ { i } μ where i = 1 , 2 , 3 , ⋯ , f [ μ ] : ( e ^ i [ μ ] , e ^ j [ μ ] ) = δ i j . The index i of the basis-vector is related to the Yamanouchi symbol M for the tableau [

Using this formula, we can look at the irreducible representations for all elements of the group. A point to note here is that not all irreducible representations may be required to describe a physical system. A good way to see which irreducible representations are relevant, we must construct a Character Table. We do so by considering the trace (character) values of the irreps for a given conjugacy class and kind of irrep (as defined by its dimensionality). The columns of the Character Tableare for elements of the same conjugacy class while the rows of the Tableare for elements of the same irrep. Once we have constructed the entire Character Table, we assign a variable (say α i for the i^{th} irrep) to each irreducible representation, and multiply these with the character of the irrep (say a i for the i^{th} irrep) for each conjugacy class, before adding them up. We equate this sum to the trace (character) of the permutation matrix for each conjugacy class ξ that we found previously:

ξ = ∑ i α i a i . (8)

This leads to a system of linear equations, whose solutions determine uniquely the kinds of irreps that are relevant along with the number of irreps of each kind that constitute the block-diagonalized form of the permutation matrices. The permutation matrices { P i , i = 1 , 2 , ⋯ , N ! } can be simultaneously diagonalized to irreps { D i , i = 1 , 2 , ⋯ , N ! } with the same block form using a similarity transformation. These two sets of matrices are related by the relation:

D i = S { − 1 } P i S , i = 1 , 2 , 3 , ⋯ , N ! (9)

Solving this set of linear equations algebraically, gives us a similarity transformation S. However, this solution is not unique and there may be several unresolved elements within the matrix representation of the similarity transformation S. For our system, we resolve this problem using the imposition of a condition of unitarity on this matrix.

Let us say we had a permutation operator P operating on an initial state | ψ 〉 to give a final output state | ψ ′ 〉 :

| ψ ′ 〉 = P | ψ 〉 . (10)

Then operating with the similarity transformation S − 1 , gives us:

S − 1 | ψ ′ 〉 = S − 1 P | ψ 〉 = ( S − 1 P S ) S − 1 | ψ 〉 = D S − 1 | ψ 〉 . (11)

Therefore, the eigenvectors of the block-diagonalized matrices are of the form:

| ϕ 〉 = S − 1 | ψ 〉 . (12)

Usually | ψ 〉 is taken to be the basis comprising of all 2 n separable n-qubit states, and the vector | ϕ 〉 is regarded as the invariant subspace vectors, since the block-diagonalized forms have blocks corresponding to the invariant subspaces of the system. So, if one was to begin with any linear combination of n-qubit vectors that are spanned by the constituent-vectors of an invariant subspace, the state that emerges out of the application of an arbitrary combination of SWAP^{α} gates is always a linear combination of those constituent-vectors of the invariant subspace as well. We find that for n-qubits, we have the following number of such vectors

N { v } = { ∑ i = 0 N 2 ( n + 1 − 2 i ) × d [ n − i , i ] if n is even ∑ i = 0 N − 1 2 ( n + 1 − 2 i ) × d [ n − i , i ] if n is odd . (13)

The vectors that comprise the invariant subspaces have characteristic entanglement patterns. Invariance of the quantum states under Stochastic Local Operations and Classical Communications (SLOCC) is a way to classify these states into entanglement-families [^{α} gate commutes with the total z-spin operator S { z } = ∑ v ∈ V σ v z , the action of the Power-of-SWAP gate U SWAP α breaks the Hilbert Space into a direct sum H ≅ ⊕ k = 0 N Γ k [

A general n-qubit state with the same Hamming weight across all its superposition states can therefore be written as

| ψ 〉 = ∑ i d a i | i 〉 (14)

where | i 〉 are the vectors associated to the invariant subspace with that Hamming weight and d denotes the dimensionality of the invariant subspace under consideration. More generally, we can express a general quantum state as

| ψ 〉 = ∑ k l ∑ i d k a i ( k ) | i ( k ) 〉 (15)

where | i ( k ) 〉 are the vectors associated to the invariant subspace with the Hamming weight k and d_{k} denotes the dimensionality of the invariant subspace with that particular Hamming weight, while l denotes the number of Hamming weights present in the superposition (system) in the quantum state under consideration.

For four-qubits states, there are 16 vector-states comprising three [

d 0 = 1 ; d 1 = 1 , 3 ; d 2 = 1 , 2 , 3 ; d 3 = 1 , 3 ; d 4 = 1 (16)

in equation (15). This is because there are multiple invariant subspaces for the same Hamming weight, with the Hamming weights 1, 2 and 3 having multiple invariant subspaces in this case. The vectors in an invariant subspace remain within that subspace given any combination of SWAP^{α}^{ }gates operated upon them.

One of the fundamental properties of the SWAP^{α} gates is that they conserve parity. The number of | 0 〉 s and | 1 〉 s in a multiqubit state remain the same. This is a key to proposing an efficient algorithm for finding the vector states associated with the invariant subspaces for the SWAP^{α} gates.

Let us say we have the following families of n-qubit states:

W i = { PERM ( | { 00 ⋯ 0 } ︸ { n − i } { 11 ⋯ 1 } ︸ { i } 〉 ) } (17)

where PERM defines the permutation function for a given set of qubits.

There is no map, f ( U SWAP α ) comprising of only SWAP^{α} that can take us from one such family W i to another family W j for i ≠ j . This property can be used to define the set of basis vector-sets by starting with a completely symmetric state using all the vectors in the family W i and then finding the null-space vectors for the same. We have found an algorithm and realization of the same that can carry out quick and efficient generation of the vector-states for n-qubits under operation by SWAP^{α} gates.

In this section, we will be looking at the accessibility of quantum states using only Power-of-SWAP gates for an arbitrary input quantum states. For the two-qubit case, let us begin with a state of the form

| ψ { i n } 〉 = α | 00 〉 + β | 01 〉 + γ | 10 〉 + δ | 11 〉 . (18)

If we have a parameter t ∈ [ 0 , 1 ] and the SWAP-gate is the unitary generated by the Heisenberg Hamiltonian for evolution time t = 1 , then we can generate the fractional SWAP by doing the Hamiltonian evolution for time t ∈ ( 0 , 1 ) . For

time steps of the form t = 1 m , m ∈ ℤ , we obtain the so-called n^{th} Power-of-SWAP (SWAP^{α}) gates.

In the two-qubit case, let’s fix a Power-of-SWAP circuit of depth m, such that for each i = 1 , ⋯ , m , we apply a U i = SWAP γ i with γ i = 1 / n i for n i ∈ ℤ . Let us look at only the subspace spanned by { | 01 〉 , | 10 〉 } . We have

| ψ i n 〉 → U m U { m − 1 } ⋯ U 1 ψ f = α ′ | 00 〉 + β ′ | 01 〉 + γ ′ | 10 〉 + δ ′ | 11 〉 , (19)

Since our unitaries are parity preserving, α ′ = α , δ ′ = δ . Now we can consider the inverse problem: fix the initial and final states, and a precision parameter ϵ ∈ ( 0 , 1 ) , and compute a sequence n 1 , ⋯ , n m such that U ( Γ ) : = U m U m − 1 ⋯ U 1 brings | ψ i n 〉 within the ϵ -ball around | ψ f 〉 , i.e.,

| | ψ f 〉 − U | ψ i n 〉 | ≤ ϵ .

Noting that U ( Γ ) = e i Γ H SWAP , we can also first solve for the time parameter Γ and then obtain the sequence { n } i . Let us do this for ϵ = 0 . Taking the | 00 〉 , | 11 〉 subspaces as invariant, we have the matrix equation

1 2 [ 1 + e i π Γ 1 − e i π Γ 1 − e i π Γ 1 + e i π Γ ] ( β γ ) = ( β ′ γ ′ ) (20)

which we can solve for Γ to get the conditions (assuming $\alpha\neq\beta$)

e i π Γ = 2 β ′ − ( β + γ ) β − γ = − 2 γ ′ − ( β + γ ) β − γ , (21)

for which to have a solution, we require β ′ + γ ′ = β + γ .

The result given above can be generalized to the case of higher-number of qubits. For any combination of Power-of-SWAP and general n-qubit input state of the form | ψ 〉 = α 0 | 000 ⋯ 00 〉 + α 1 | 000 ⋯ 01 〉 + ⋯ + α 2 N | 111 ⋯ 11 〉 being transformed to | ψ ′ 〉 = β 0 | 000 ⋯ 00 〉 + β 1 | 000 ⋯ 01 〉 + ⋯ + β 2 N | 111 ⋯ 11 〉 . We derive a condition for the coefficients of the vector basis states with the same Hamming weights. Let denote the set of all coefficients associated with vector states with

the same Hamming weight i as { α ( h w ) ( i ) k } in the input quantum state and denote the set of all coefficients associated with vector states with the same Hamming weight { β ( h w ) ( i ) k ′ } in the output quantum state. Here k and k' denote the indices of the coefficients in each such set. Then,

∑ k α ( h w ) ( i ) k = ∑ k ′ β ( h w ) ( i ) k ′ (22)

for any n-qubit case. This is a powerful result since it, along with the normalization condition, helps us determine the kinds of states that are derivable from the operation of just Power-of-SWAP gates on a general n-qubit state.

In this paper, I have found a way to generate all the basis vector-states that span the space of states accessible by an arbitrary Power-of-SWAP gate. In doing so, we have found several classes of states. These include maximally entangled states such as the W-states, partially entangled cluster states and separable states. Each of these can be used for various kinds of applications in quantum information processing.

With the exponential scaling of the computation basis with the number of basis, we see that the basis vectors of the invariant subspaces also increase exponentially. It was found that only specific states can be generated using an initial quantum state and a combination of SWAP^{α} gates, and hopping between these points represents a quantum computation. By understanding the invariant subspaces, I realize an accessible resource for quantum computation using SWAP^{α}. This quantum computation is resilient to parity changes and other forms of errors due to the permutation symmetries of the vectors comprising the invariant subspaces. I hope that this paper will pave the way for the use of the SWAP^{α} for generation of quantum resource-states in quantum information processing tasks.

I would like to acknowledge the contribution of Trinity College, Cambridge and the Nehru Trust for Cambridge University (NTCU) who funded the project. I would like to acknowledge the contributions of Prof. Crispin Barnes, Mr. Yordan Yardanov, Dr. David R. M. Arvidsson-Shukur and Mr. Sathyageeshwar Subramaniam.

The author declares no conflicts of interest regarding the publication of this paper.

Majumdar, M.G. (2019) Generation of Quantum Vector States Using the Exchange Interaction. Journal of Applied Mathematics and Physics, 7, 3116-3131. https://doi.org/10.4236/jamp.2019.712219