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Quantum Fourier Transform in a Decoherence-Free Subspace

Quantum Fourier transform is of primary importance in many quantum algorithms. In order to eliminate the destructive effects of decoherence induced by couplings between the quantum system and its environment, we propose a robust scheme for quantum Fourier transform over the intrinsic decoherence-free subspaces. The scheme is then applied to the circuit design of quantum Fourier transform over quantum networks under collective decoherence. The encoding efficiency and possible improvements are also discussed.


I. INTRODUCTION
Quantum Fourier transform (QFT) plays essential roles in various quantum algorithms such as Shor's algorithms [1,2,3] and hidden subgroup problems [2,4,5]. Inspired by the exponential speed-up of Shor's polynomial algorithm for factorization [1], many people investigated the problem of efficient realization of QFT in a quantum computer [3,6,7,8,9]. Up to now, many improvements have been made. In [6], Moore and Nilsson showed that QFT can be parallelized to linear depth in a quantum network, and upper bound of the circuit depth was obtained by Cleve and Watrous [7] for computing QFT with a fixed error. In [8] the actual time-cost for performing QFT in the quantum network was examined. Further, Blais [9] designed an optimized quantum network with respect to time-cost for QFT.
In practice, the decoherence problem induced by the unavoidable coupling of quantum system with the environment have to be considered in circuit design for QFT over a quantum network. If no measure is taken, decoherence will destroy the encoded quantum information. Many methods have been proposed to suppress decoherence in a quantum system, among which, an important scheme is to encode the quantum information into the decoherence-free subspaces or subsystems (DFS) [10,11,12,13,14,15,16,17,18] of quantum system. Theoretically, DFSs are completely isolated from the noises [10,11,12,13]. A large amount of discussions about DFS have appeared in the literature [14,15,16,17,18].
In this paper, we will take advantage of the decoherence-free subspaces to develop a novel scheme for performing QFT in a quantum computer. The circuits designed in this way have the robustness against noise in the procedure of implementing QFT. The paper is orga-nized as follows: in Sec.II, some notations and the preliminary knowledge on DFS and QFT will be reviewed; in Sec.III, general method will be introduced for implementing QFT in DFS; in sec.IV, circuits will be designed to perform QFT in the DFS of a quantum network with respect to weak collective decoherence (WCD) and strong collective decoherence (SCD) respectively; in Sec.V, the efficiency of the circuit and possible improvements will be discussed; finally, a conclusion will be made in Sec.VI.

II. NOTATIONS AND PRELIMINARIES
Suppose the quantum system S under consideration is coupled to an environment E. The overall system is governed by the Hamiltonian in the form of [16]: where S α (E α ), (α ∈ Λ) are operators acting on the state space H S (H E ) of S(E), and the index set Λ contains all the possible couplings between the system and the environment. Assume S α (α ∈ Λ) span a †-closed associate algebra A. According to [19], A is isomorphic to a direct sum of d J × d J complex matrix algebras, each with multiplicity n J [15,16] where the index set J labels all the irreducible components of A. Correspondingly, the system Hilbert space H S can be decomposed into a similar form All the subsystem spaces C nJ (J ∈ J ) in the right hand side of Eq.(3) correspond to decoherence-free subsystems of the quantum system S. Particularly, C nJ gives a decoherence-free subspace of the quantum system S when d J = 1.
Quantum network under collective decoherence (CD) provides a nice paradigm for the DFSs. Roughly speaking, all qubits of a quantum network under CD are coupled to the environment in the same manner. In the literature [16], two types of CD, weak collective decoherence (WCD) and strong collective decoherence (SCD), are frequently discussed.
SCD is defined as the decoherence due to the interaction Hamiltonian where S α = n i=1 σ i α , and σ i α (α = x, y, z) represents the Pauli matrix σ α that corresponds to the local operation on the i th qubit.
If only one term appears in the right hand side of Eq.(4), i.e. the system is coupled to the environment only in one direction, the induced decoherence is called WCD. Without loss of generality, the Hamiltonian can be written as Next, we give a brief description of QFT implemented over an n-qubit quantum network. Mathematically, the quantum Fourier transformation QF n can be expressed as [9]: Denote the state of the quantum network by the qubit string |s n s n−1 · · · s 2 s 1 , (s t ∈ {0, 1}, t = 1, 2, · · · , n) in which the t th qubit is at the state |s t . The transformation QF n can be realized by applying the following sequence of quantum gates (all the gate sequences in this paper are operated from the right to left one by one) where )H (k) .
over k th qubit. The controlled-phase-shift gate P (i,j) (θ) represents the action over i th (control) and j th (target) qubits.
In addition, there are three important elementary gates that will be used in this paper. The controlled-NOT gate CN (i,j) flips the j th qubit (target qubit) when the i th qubit (control qubit) is at the state |1 , and nothing is done when the control qubit is at the state |0 . The rotation gate R (k) (α) realizes the unitary transformation over k th qubit: The controlled-rotation gate CR (i,j) (β) realizes the rotation R (j) (β) over j th qubit (target qubit) when the state on the i th qubit (control qubit) is |1 , and nothing is done when the state on the i th qubit (control qubit) is |0 . More details about DFS and QFT can be found in ref. [3,9,15,16] and the references therein.

III. SCHEME FOR PERFORMING QFT IN A DFS
In the following parts of this paper, we focus the study on implementing QFT over decoherence-free subspaces. If not claimed, the abbreviation DFS will indicate only the decoherence-free subspace. Suppose H DF S is an n J dimensional DFS of the quantum system S, then one can select 2 [log 2 nJ ] orthonormal states in H DF S to construct at most [log 2 n J ] new qubits. For clarity, we call these qubits logical-qubits, and the original qubits physicalqubits.
Next, we will discuss how to realize a robust QFT algorithm over these logical-qubits. Similar to (6), we define the QFT over a DFS by where the states |φ = |ŝ nŝn−1 · · ·ŝ 1 ,ŝ t ∈ {0,1}, t = 1, 2, · · · , n are basis states of the logical-qubits. The basic idea for realizing QF n(DF S) is as follows. Notice that the two classes of gates H (k) and P (i,j) (θ) play the central roles in the QFT [3,7,8,9], we will construct correspondingly two similar classes of quantum gates for implementing QFT in a DFS, denoted by onelogical-qubit gate H (k) DF S acting on the k th logical-qubit and two-logical-qubit gate P (i,j) DF S (θ) acting on the i th and j th logical-qubits respectively. These two classes of gates should fulfil two requirements: one is that they are invariant operators on the state space of the logical-qubits; the other is that they operate on the logical-qubits in the same manner as H (k) and P (i,j) (θ) on the physicalqubits.
Similar to the general QFT described in Eq.(6), we can realize QF n (DF S) by applying the following sequence of quantum gates where T (k) Gate sequence (13) provides us the general strategy for designing a circuit to implement QFT over the DFS in a quantum system.
Similarly, we have and From Eqs. (19-21), we can see that if U can be constructed by elementary gates, then U −1 H (k) U and U −1 P (i,j) U are feasible realizations for the two gates H The remainder tasks, then, are to find the transformation U in Eq. (15) and build a circuit to realize it. From the theory of universal quantum computation [20], any unitary operator can be constructed by a sequence of universal elementary gates. In most cases it is not easy to obtain such explicit decompositions. Whereas, as will be shown in the next section, it is possible to build up a circuit for QF n(DF S) over the quantum network under collective decoherence with a finite number of elementary gates.

IV. CIRCUITS FOR QFT OVER QUANTUM NETWORKS UNDER COLLECTIVE DECOHERENCE
A. The weak collective decoherence case In the quantum networks under WCD, nontrivial DFS exists only when the original network has no less than two physical-qubits [16]. For the simplest case, the DFS in a two-qubit quantum network under WCD is spanned by the orthonormal states |01 and |10 [16], with which one can build up one logical-qubit, i.e. and For a 2n-qubit quantum network under WCD, we use the orthonormal states |ŝ n ⊗ |ŝ n−1 ⊗ · · · ⊗ |ŝ 1 , (ŝ t ∈ {0,1}, t = 1, 2, · · · , n), where |ŝ t represents the t th logical-qubit extracted from the (2t − 1) th and (2t) th physical-qubits, to construct the circuit for robust QFT. It can be verified that all these states are contained in the biggest DFS.
Over these logical-qubits, it is observed that H (k) DF S and P (i,j) DF S (θ) can be directly constructed from a sequence of elementary gates as follows (the circuits are given in Fig.1 and Fig.2): Being able to perform the gates H Let U n = CN (2,1) CN (4,3) · · · CN (2n,2n−1) . Observing that the term CN (2t,2t−1) commutes with H (2k) when t = k and commutes with P (2i,2j) (θ) when t = i or j, we have and Therefore, we can choose U n as the unitary transformation U in Eq. (15): Consider the three-qubit QFT as a simple example, the transformation QF 3(DF S) can be realized by applying DF S (θ) are those given in Fig.1 and Fig.2.

B. The strong collective decoherence case
It is more complicated to design the circuit for QFT over quantum networks under SCD than WCD. The corresponding condition for the existence of a DFS is more critical. Quantum network with four physical-qubits is of the smallest scale to ensure the existence of a nontrivial DFS, which is spanned by two orthonormal states [16] and Naturally, |0 and |1 form one logical-qubit. By dividing the physical qubits into 4-qubit units, one can use the canonical basis |s n ⊗ |s n−1 ⊗ · · · ⊗ |s 1 , (s t ∈ {0,1}, t = 1, 2, · · · , n) , where |s t represents the t th logical-qubit extracted from the (4t − 3) th to (4t) th physical-qubits, to construct n logical-qubits in a 4n-physical-qubit quantum network under SCD.
To perform QFT over the logical-qubits obtained above, it is still crucial to design the circuits for the corresponding two classes of gates H Let U (k) be an unitary transformation on the physicalqubits from (4k − 3) th to (4k) th , which is realized by applying the sequence of elementary gates as follows (see the circuits for the transformation U (k) and its inverse in Fig.4): where α = π − arcsin 1 √ 3 , β 1 = −π + arcsin 1 √ 3 , β 2 = − π 4 . Then, in a 4n-physical-qubit quantum network, one of the feasible realization of the transformation U is: With the help of the unitary transformation U , the fundamental gates H where the gates H DF S (θ) satisfy the requirements given section III: The circuit for performing QFT in the DFS of quantum network is constructed by substituting the operators H (k) DF S and P (i,j) DF S (θ) into the gate sequence in (13).

V. THE EFFICIENCY AND OPTIMIZATION
The encoding efficiency of quantum algorithms over the DFS of an n-qubit quantum network, say η(n), is defined as the ratio of the number of logical-qubits to that of physical-qubits. The efficiency depends on the selection of DFS and the way of building logical-qubits. From section III, it is obvious that the maximum encoding efficiency is: It has been derived in [12,16] that the efficiency η max (n) of the quantum network under collective decoherence approaches to 1 when n → ∞. For the circuits we designed for QFT over the quantum network under collective decoherence, the encoding efficiency η(n) = 1 2 for WCD and η(n) = 1 4 for SCD. Therefore, it is possible to design a more efficient circuit for realizing QFT over the DFS of some quantum network under collective decoherence. However, our circuits are scalable for they are relatively easy to be realized for large scale robust QFT over quantum networks. Consequently, there is a trade-off between the encoding efficiency and circuit complexity.
For example, if we want to implement m-qubit QFT in a DFS of some quantum network under collective decoherence, then at least physical-qubits are required. The corresponding circuit for QFT over this r-qubit quantum network is the most efficient, but it will become much more complicated in using more elementary gates. The circuit design will be a formidable task.

VI. CONCLUSION
In this paper, strategies for performing QFT in a quantum network coupled with the environment are discussed. We propose a scheme for noise-isolated QFT over the decoherence-free subspaces. Following the scheme, circuits for implementing QFT are designed in quantum network under collective decoherence. The efficiency of these circuits and some possible improvements are discussed as well.
In the future, a general designing methodology needs to be found for more efficient QFT over arbitrary quantum network. Also, it is worthwhile to reduce the number of elementary gates using in the relevant quantum circuits. Moreover, it is interesting and useful to extend the problem from the decoherence-free subspaces to decoherencefree subsystems.