Generation of Quantum Vector States Using the Exchange Interaction

In this paper, I study the invariant subspaces of quantum states under SWAPα 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 SWAPα gates, given an initial input state.


Introduction
The non-local physical phenomena related to quantum entanglement [1] [2] [3] have played a pivotal role in the development of quantum mechanics during the last century. Entanglement in many-particle states has been extensively studied [4] [5] [6], and such states are known to have symmetries that help in determining their entanglement properties [7] [8]. This is useful in determining the usability of these quantum states as resource-states for application in quantum information processing tasks [9]- [14]. Prior to using these resource-states, the generation of these quantum states is important, and this is an arduous task that requires a high degree of control. Quantum state engineering has been a way to generate arbitrary quantum states experimentally [15]- [19]. The existence of entanglement equivalence classes [20] of quantum states, particularly symmetric states, simplifies the experimental generation of these states. By producing a single state of an equivalence class, the entire subspace of states in that class can be accessed using a reduced set of operations. As a result of this, the characterization of symmetries in quantum states goes conjointly with the study of re-certain physical systems, some of which have been recently implemented experimentally [29] [30] [31]. A (partial) permutation on two qubits can be carried out by a (partial) SWAP gate [32]. Moreover, the SWAP α gate, together with single-qubit operations, is found to produce a universal set of quantum gates (enabling any quantum computation) [33]. These two-qubit gates can be realised in several physical systems. In electronic systems, the exchange interaction provides a natural framework for the implementation of the SWAP α gate [34] [35], but equivalent implementations also exist for atomic systems [36] [37], nitrogen vacancies [38] and photonic systems [39] [40] [41] [42].
Burkard et al. [43] showed how the SWAP gate can be used to create a controlled phase-flip gate (CPHASE), which in turn can be used to create a XOR gate. Barenco et al. [44] showed that any unitary two-qubit gate can be simulated by four one-bit gates and two XOR gates, thereby effectively leading to universal quantum computation using only the SWAP and single qubit rotation gates}. Since local single-qubit operations are difficult to implement in hardware, Divincenzo et al. subsequently looked at how universal quantum computation can be achieved by using only SWAP α gates if one encodes pseudo-spin qubits using three physical qubits [33]. This naturally leads to the question of whether universal quantum computation could be implemented purely using SWAP α with a specific value of α. It was shown by Tanamoto et al. [45] that upon preparing the initial state as ( )( ) 1 0 1 0 1 2 ± = + − and applying the SWAP one can obtain Raussendorf's cluster state [46] but only after two single-qubit rotations.
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 n S 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 Journal of Applied Mathematics and Physics 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.

SWAP α and Invariant Subspaces
Quantum logic gates are essential building blocks of a quantum computer. The SWAP α quantum gate with 0 1 α < ≤ 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 [47] [48] [49]. The SWAP α gate can be experimentally implemented in several physical systems such as in the exchange interaction between electrons trapped by surface acoustic waves [34] [50]. In our paper, we look at the use of only SWAP 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.   [22] cycles that correspond to a 12 34 T T -invariant and 13 24 T T -invariant subspaces respectively, where ij T correspond to transposition of the i th and j th qubit.
All the vector states that are associated with any specific invariant subspace corresponding to the Symmetric Group n S are found to have the same Hamming weight in their qubit representation. We find that the number of basis 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 2 n k ≤ , we find that the relevant partitions (and corresponding permuta- we consider all possible values of k for a fixed value of n, the total number of sets of vector-states that correspond to the partition [ ] n is ( )  N , for an n-qubit system, operated upon by the SWAP α , is given by: qubits in between; rotating the first three qubits among themselves, while rotating the next three qubits among themselves, is equivalent to rotating all the qubits (to the right) once; (b) the 0 qubits have one 1 qubit in between, as shown; then carrying out the same operation: rotating the first three qubits among themselves, while rotating the next three qubits among themselves, is equivalent to rotating by a [5,1] transposition followed Journal of Applied Mathematics and Physics by two instances of the complete rotation [6]. (2) Invariant subspaces of the Symmetric Group n S and reducibility of representations are closely linked [51]. If a representation ρ of the group G has no non-trivial invariant subspace, we say that it is irreducible. Defining as the SWAP operation between the i th and j th qubits in an n-qubit state can be transformed into a block form D, comprising of irreducible representations of the Symmetric Group n S 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: This similarity transformation is found to be useful in block-diagonalizing the basic SWAP α -gates representations as well, since by definition, where ( ) 2 2 I × is a 2 2 × identity matrix. As a result, the similarity transformation S that transforms the composite SWAP operator to a block-diagonal form: The matrix decomposition thus formed is block diagonal, as shown in the illustrative example in Figure 2 for the case of two matrices. Thus, we find that the various combinations of n copies of SWAP α gates, operating over a multi-qubit input state, produce a group that is isomorphic to the symmetric group n S . In fact, any function of SWAP α gates can be put into the block form using the same similarity transformation:  showing that various combinations of n copies of SWAP α gates, operating over a multi-qubit input state, produce a group that is isomorphic to the symmetric group n S .

Generation of Vector States
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: 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 n S .
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 → of the group. The first step in this regard is to find the dimensions of the irreps using the Young's Tableau [53]. For any n qubit system, we can find the dimensionality of the irrep using the hook lengths of the elements in it [54] [55] and the hook product [56] h of the tableau. The dimension d of the irrep is then given by 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 [57]  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 Table are for elements of the same conjugacy class while the rows of the Table are for elements of the same irrep. Once we have constructed the entire Character Table, we assign a variable (say i α for the i th irrep) to each irreducible representation, and multiply these with the character of the irrep (say i a 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: 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 Solving this set of linear equations algebraically, gives us a similarity transformation S. However, this solution is not unique and there may be several un-Journal of Applied Mathematics and Physics resolved 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 ′ : Then operating with the similarity transformation 1 S − , gives us: Therefore, the eigenvectors of the block-diagonalized matrices are of the form: 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 nqubit 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

Symmetry under Parity and Efficient Algorithm for Generating n-Qubit Vector States for SWAP α Gates
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: where PERM defines the permutation function for a given set of qubits.
There is no map,

Accessibility of Quantum States Using Power-of-SWAP Gates
In this section, we will be looking at the accessibility of quantum states using only which we can solve for Γ to get the conditions (assuming $\alpha\neq\beta$) ( ) ( ) 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.

Conclusions
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.