2 σ x I 2 × 2 + + I 2 × 2 I 2 × 2 σ x (31)

S y = σ y I 2 × 2 I 2 × 2 + I 2 × 2 σ y I 2 × 2 + + I 2 × 2 I 2 × 2 σ y (32)

S z = σ z I 2 × 2 I 2 × 2 + I 2 × 2 σ z I 2 × 2 + + I 2 × 2 I 2 × 2 σ z (33)

For the condition S α | ψ = a α | ψ , a α to hold true for each of these forms of S, we must have states that will give a global and not local phase across the superposition in the operators S x , S y and S z . This is only possible if | ψ = | + | + | + or | | | for the operator of the form in Equation (31), | ψ = | + y | + y | + y or | y | y | y for the operator of the form in Equation (32), and | ψ = | 0 y | 0 y | 0 y or | 1 y | 1 y | 1 y for the operator of the form in Equation (33), where | ± = 1 2 ( | 0 ± | 1 ) and | ± y = 1 2 ( | 0 ± i | 1 ) . These can never be simultaneously true. As a result,

S α | ψ = 0 (34)

Exponentiating the operators in (30) and using (34), we have

O ( p ) = exp [ α ( S α a α I ) p α ] | ψ = exp [ α ( a α I ) p α ] | ψ (35)

If we now consider the hamiltonian for the exchange interaction: E | a | b | b | a , we see that

O ( p ) E = exp [ α ( a α I ) p α ] E = E × exp [ α ( a α I ) p α ] = E × O ( p ) (36)

Thus, given the result in Equation (31), the operator given by the exchange interaction preserves the decoherence-free subspace for a “collective deco-herence” model. The smallest number of physical qubits that gives a fully encoded Decoherence-free Subspace qubit is found to be four [71]. Let us take this case, and consider the states with zero angular momentum:

| 0 L = 1 2 ( | 01 | 10 ) ( | 01 | 10 ) (37)

| 1 L = 1 3 ( | 00 | 11 1 2 ( | 01 + | 10 ) ( | 01 + | 10 ) + | 11 | 00 ) (38)

Let us now look at the effect of the operation of the various exchange interactions E i j , where the ith and jth qubits are being exchanged.

E 12 | 0 L | 0 L , E 12 | 1 L | 1 L (39)

Due to the symmetry of the logical basis states, E 34 has the same effect. Looking at the operation, we can define an encoded Z ¯ operator:

Z ¯ = E 12 = E 34 (40)

For defining a similar X ¯ operator: X ¯ | 0 L | 1 L , X ¯ | 1 L | 0 L is not as straightforward since no one exchange interaction seems to provide the solution. Therefore, before moving forward with trying to define this composite operator, let us look at some other cases for the exchange interaction:

E 13 | 0 L 1 2 ( | 0101 | 1100 | 0011 + | 1010 ) = 1 2 | 0 L 3 2 | 1 L ,

E 13 | 1 L 1 3 ( | 1001 1 2 ( | 0101 + | 1100 + | 0011 + | 1010 ) + | 0110 ) = 3 2 | 0 L 1 2 | 1 L (41)

Again, due to the symmetry of the states, the case for E 24 gives the same results. Using operators E 12 and E 13 (or E 34 and E 24 ), we can define the X ¯ operator

X ¯ = 1 3 E 12 2 3 E 13 = 1 3 E 34 2 3 E 24 (42)

The ability to implement these primary logical operations is sufficient to implement any gate in S U ( 2 ) on the encoded qubits, by using the Euler angle reconstruction (about any two orthogonal axes):

exp ( i ω ( n σ / 2 ) ) = exp ( i β σ z / 2 ) exp ( i θ σ y / 2 ) exp ( i α σ z / 2 ) (43)

the resulting rotation is given by the angle ω about the direction specified by the unit vector n, both of which are functions of α , β and θ . Mapping ( σ x , σ y , σ z ) ( X ¯ , Y ¯ , Z ¯ ) , we can construct any element of SU(2) in the encoded space by turning on and off the appropriate exchange interaction.

For two qubit gates, we have to construct slightly more complex combinations of gates. Let us start with the controlled-Phase shift gate (CPHASE). The idea is to introduce a phase for the last case and not for any of the others. With some clever usage of the exchange interactions, this can be done:

( E 12 E 56 ) ( E 12 E 56 2 I ) (44)

This gives us a phase only for the case for | 11 L . The CPHASE gate has been previously realized in a different manner by Bacon et al. [71] using the operators: h 1 = [ E 26 , E 12 + E 25 ] + [ E 15 , E 12 + E 16 ] , h 2 = j = 5 8 ( E 1 j + E 2 j ) and

c = 1 32 [ h 1 , ( h 2 , h 1 ) ] . As can be seen, our operator is a lot simpler in construction.

The CNOT gate can be realized similarly using two logical qubits. We find the form of this operator in the encoded space to be

1 3 ( I E 12 ) ( E 56 2 E 57 ) (45)

The CNOT gate has been realized previously with two logical qubits comprising of three physical qubits as well by DiVincenzo et al. [26].

Thus, one can obtain a fault-tolerant universal set of gates using just the exchange interaction.

6. Discussion

Since the S W A P α gates generated by the exchange interaction are non-commutative in general (adjacent operators operating on a common qubit), we can obtain an entire zoo of quantum states (and entanglement patterns) using these operators. However, given the parity constraints due to the permutation symmetries involved, we can define invariant subspaces for these states. This gives us an efficient method for classifying the generated states.

This resource and classification has been used for defining various models of quantum computation, quantum communication, quantum memory and deo-cherence-free subspaces. The Hamming-weight preserving symmetry of the S W A P α gate is found to be of fundamental importance in most of these protocols. Due to the ubiquity of the exchange interaction in various physical systems including in condensed matter systems, this set of applications can be extremely useful in realizing quantum information processing in such physical systems.

7. Conclusion

In this paper, I have proposed new ways of applying entanglement generated using the exchange interaction for various quantum information processing tasks. This includes five distinct models of quantum computation, ways of implementing quantum communication protocols, quantum memory and defining decoherence-free subspaces. Future experimental realizations of quantum information processing that involve the exchange interaction can be based on the comprehensive theoretical study of these applications given in the paper.

Acknowledgements

We 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 Professor Crispin Barnes for helping with this paper.

Cite this paper
Majumdar, M. (2018) Quantum Information Processing Using the Exchange Interaction. Journal of Quantum Information Science, 8, 139-160. doi: 10.4236/jqis.2018.84010.
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