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The Bell states preparation circuit is a basic circuit required in quantum teleportation. We describe how to implement it in all-fiber technology. The basic building blocks for its implementation are directional couplers and highly nonlinear optical fiber (HNLF). Because the quantum information processing is based on delicate superposition states, it is sensitive to quantum errors. In order to enable fault-tolerant quantum computing the use of quantum error correction is unavoidable. We show how to implement in all-fiber technology encoders and decoders for sparse-graph quantum codes, and provide an illustrative example to demonstrate this implementation. We also show that arbitrary set of universal quantum gates can be implemented based on directional couplers and HNLFs.
©2010 Optical Society of America
1. Introduction
Quantum information processing (QIP) is an exciting research area with numerous applications including quantum key distribution (QKD) and teleportation, quantum computing, quantum lithography and quantum memories [1], [2]. To perform an arbitrary quantum computation, a minimum number of gates known as universal quantum gates [1–4] is needed. The most popular sets of universal quantum gates are: {Hadamard (H), phase (S), CNOT, Toffoli (UT)} gates, {H, S, π/8 (T), CNOT} gates, Barenco gate [2] and Deutsch gate [4]. Because quantum effects are easily observable in optical systems, the QIP by photons represent one of the earliest proposal for quantum gates implementation [5–7]. For instance, it has been shown by Knill, Lafflame and Milburn [6] that efficient quantum computation/teleportation is possible with linear optics, but requires the use of 2n modes for single-qubit teleportation. The most of proposals for various gates implementation with linear optics [6–8] provide only probabilistic outcomes, and as such are not suitable for large-scale computation (in the order of thousand and above). On the other hand, various proposals based on Kerr nonlinear effect, e.g [5], require the use of bulky devices.
Optical fibers are useful for low-power all-optical signal processing and QIP. For example, the authors in [2] described an all-fiber implementation of the Deutsch-Jozsa and Bernstein-Vazirani quantum algorithms with the measured visibility of the corresponding 8-path interferometer of about 97.5%. Moreover, it has recently been demonstrated by Matsuda et al. [9] that Kerr nonlinearity can introduce the cross-phase modulation phase shift at single-photon level by using photonic crystal fibers. Inspired by this recent achievement, in this paper we show that arbitrary family universal quantum gates can be implemented in all-fiber technology based on directional couplers and highly nonlinear optical fibers (HNLFs) by employing photon polarization. The key difference between our proposal and previous proposals [6–8] is that our proposal employs the photon polarization, simple directional couplers and HNLFs to perform any quantum computation operation.
The photon polarization is fragile resource for quantum computation and teleportation, and it is very much sensitive to various polarization effects in fibers. To deal with quantum errors, the quantum error correction coding (QECC) appears to be the most efficient approach [1], [9–12]. Among different candidates for QECC, the sparse-graph quantum codes [11,12] seem to be the most promising thanks to the sparseness of their quantum-check matrices so that small number of interactions per stabilizer are needed. The basic gates needed for quantum LDPC encoding and decoding are Pauli gates, which can be implemented using all-optical fiber technology. The entanglement EPR preparation, needed in QKD and quantum teleportation, can also be implemented in the same technology, as shown later in the text. Finally, we discuss the implementation of two important class of codes, dual-containing and entanglement-assisted, in all-fiber technology.
The paper is organized as follows. In Section 2, we describe how to implement in all-fiber technology four sets of universal quantum gates, listed above. In the same section, we describe Pauli gates implementation using the same technology. In Section 3, we describe the all-fiber implementation of EPR pair preparation circuit. The implementation of photonic QECC encoders and decoders in all-fiber technology is discussed in Section 4. We restrict our attention to sparse-graph quantum codes [11,12] because their quantum check matrices are sparse so that the small number of interaction is needed in corresponding syndrome operators. We compare performance of two classes of sparse-graph quantum codes, dual-containing [11,12] and entanglement-assisted (EA) [13], and show that EA sparse-graph quantum codes significantly outperform corresponding dual-containing codes.
2. Universal quantum gates and Pauli gates in all-fiber technology
In this section, we describe all-fiber implementation of four sets of quantum gates: (i) Barenco gate, (ii) Deutsch gate, (iii) {H, S, CNOT, UT} gates, and (iv) {H, S, π/8, CNOT} gates. A set of quantum gates U s is said to be universal if it contains a minimum finite number of gates so that arbitrary unitary operation can be performed with arbitrary small error probability by using only gates from that set. The Gottesman-Knill theorem [1] showed that gates from Clifford group U C = {H,S,CNOT} are not sufficient to perform arbitrary quantum operation. If, however, the set U C get extended by either UT or π/8 gate, we obtain a universal set of quantum gates. We turn our attention now to the integrated optics implementation of four sets of universal quantum gates, by using directional couplers and HNLF as basic building blocks. In what follows, the logical “0” is represented by a horizontal (H) photon |H〉≡|0〉 = (1 0)T and the logical “1” is represented by a vertical (V) photon |V〉≡|1〉 = (0 1)T.
An arbitrary single qubit gate can be implemented in all-fiber technology as shown in Fig. 1 . We use a polarization beam splitter (PBS) at the input of quantum gate and a polarization beam combiner (PBC) at the output of the gate. The horizontal output (input) of PBS (PBC) is denoted by H, while the vertical output (input) of PBS (PBC) is denoted by V. The input qubit is denoted by |ψ〉 = ψH|0〉 + ψV|1〉 = [ψH ψV]T, while the output qubit is denoted by |ψo〉 = ψo,H|0〉 + ψo,V|1〉 = [ψo,H ψo,V]T. In Fig. 1(a), we show an implementation based on Barenco-type proposal [2]. It can be shown that output qubit is related to input qubit by:
In Eq. (1) θ = kl, where k is the coupling coefficient and l is the coupling region length. By setting appropriately phase trimmers voltages, we can perform the arbitrary single-qubit operation. For example, by setting α = ϕ = 0 rad and θ = 2π we obtain an identity gate; while by setting α = 0 rad, ϕ = π and θ = π/2 we obtain the X-gate. In Fig. 1(b), we show an implementation based on Y-Z decomposition theorem [1]. The output qubit is related to the input qubit by:
Fig. 1 Integrated optics implementation of arbitrary single-qubit gate based on single directional coupler: (a) Barenco-type gate, and (b) Z-Y decomposition theorem based type. PBS/C: polarization beam splitter/combiner.
In Eq. (2) γ = 2kl. By setting α = π/4, β = π/2, γ = 2π, and δ = 0 rad, the U-gate described by (2) operates as the phase gate:
By setting α = π/8, β = π/4 rad, γ = 2π and δ = 0 rad, the U-gate operates as π/8 gate:
Finally, by setting α = π/2, β = 0 rad, γ = π/2 and δ = π, the U-gate given by Eq. (2) operates as Hadamard gate:To complete the implementation of set (iv) of universal quantum gates, the implementation of CNOT-gate is needed. The authors in [7], [8] proposed the use of directional couplers to implement the CNOT-gate. For the completeness of presentation, in Fig. 2(a) we provide a simplified version of CNOT-gate proposed in [7], [8]. We see that the control output qubit [c H,o,c V,o]T is related to the input control qubit [c H,c V]T and input target qubit (t H,t V)T by [7]Because the output control qubit is affected by input target qubit, the definition of CNOT-gate operation (control qubit must be unaffected by target qubit) is violated [1]. This gate operates correctly only with probability of 1/9, and is essentially a probabilistic gate. In Fig. 2(b), we show the deterministic implementation of CNOT-gate based on single-qubit gate shown in Fig. 1 and HNLF. Notice that different implementations of CNOT-gate in [5,7,8] employ multi-rail/dual-rail representation, while our implementation is based on single photon polarization (spin angular momentum). Because different quantum gates implementations in [5,7,8] require either two modes for dual-rail representation [7,8] or 2n-modes for multi-rail representation [5] they cannot be implemented at all using conventional single mode fibers (SMFs). This is a key disadvantage of multi-rail representation compared to our proposal. It can be shown that output control |C o〉 = [c H,o c V,o]T and target qubits |T o〉 = [t H,o t V,o]T are related to corresponding input qubits by:The Kerr nonlinearity device in Fig. 2(b) performs the controlled-Z operation. In the absence of control c v-photon, the target qubit is unaffected because H 2 = I (identity operator). In the presence of control c V-photon, thanks to the cross-phase modulation in HNLF, the target vertical photon experience the phase shift χL, where χ is the third order nonlinearity susceptibility coefficient and L is the HNLF length. By selecting appropriately the fiber length we obtain χL = π and the overall action on target qubit is HZH = X, which corresponds to the CNOT gate action.Fig. 2 All-fiber implementation of CNOT gate: (a) probabilistic gate proposed in [7] and (b) deterministic gate proposed here.
The Toffoli gate can straightforwardly be obtained as generalization of the CNOT gate above by adding additional control qubit; while Deutsch gate can be obtained by employing three control qubits, instead of one used in (7).
In the rest of this section, we describe the implementation of Pauli gates X, Y, and Z in integrated optics based on single-qubit gate shown in Fig. 1(b). By appropriately setting the phase shifts of U-gate α, β, γ, and δ, we can obtain the corresponding Pauli gates. The Y-gate is obtained by setting α = π/2, β = δ = 0 rad and γ = π:
The Z-gate is obtained by setting α = π/2, β = π, γ = 2π, and δ = 0 rad:Finally, the X-gate is obtained by setting α = π/2, β = -π, γ = π, and δ = 0 rad:3. Implementation of Bell states (EPR pairs) preparation circuit in all-fiber technology
In this section, we describe the implementation of EPR pairs (Bell states) preparation circuit in all-fiber technology, required in quantum teleportation systems, which is shown in Fig. 3 . Among many possible versions of Hadamard and CNOT-gates we have chosen two with similar propagation times. The upper circuit operates as Hadamard gate, while the rest of the circuit operates as CNOT gate as already explained in description of Fig. 2(b).
It can be shown that output quantum state |EPRij〉 is related to the input state |ψin〉 by:
For example, by setting c H = t H = 1 and c V = t V = 0 we obtain the Bell state4. Photonic QECC encoders and decoders
In this section, we describe two classes of sparse-graph quantum codes: (i) quantum dual containing low-density parity check (LDPC) codes and (ii) entanglement-assisted LDPC codes and show that corresponding encoders and decoders can be implemented in integrated optics. Most practical quantum codes belong to the class of CSS codes [1], and can be designed using a pair of conventional linear codes satisfying the twisted property (one code includes the dual of another code). The CSS codes based on dual-containing codes are simplest to implement. Their (quantum) check matrix can be represented by
where HH T = 0, which is equivalent to C ⊥(H)⊂C(H), where C(H) is the code having H as the parity check matrix, and C ⊥(H) is its corresponding dual code. The quantum LDPC codes have many advantages over other classes of quantum codes, thanks to the spareness of their parity-check matrices [11,12]. From Eq. (12) it follows that by providing that the H-matrix of a dual-containing code is sparse, the corresponding A-matrix will be sparse as well, while corresponding stabilizers will be of low weight. For example, the H-matrix given below satisfies the condition HH T = 0, and can be used in quantum check matrix (12) as dual containing code:The main drawback of dual-containing LDPC codes is the fact that they are essentially girth-4 codes*, which do not perform well under sum-product algorithm (commonly used in decoding of LDPC codes). On the other hand, it was shown in [13] that through the use of entanglement arbitrary classical codes can be used in correction of quantum errors, not only girth-4 codes. Because QKD and quantum teleportation systems assume the use of entanglement, this approach does not increase the complexity of the system at all. The number of entanglement qubits (ebits) needed in EA LDPC codes is e = rank(HH T) (where H is the parity-check matrix of an LDPC code and rank(.) is the rank of a given matrix) so that minimum number of required EPR pairs is one, exactly the same as already in use in QKD scheme. For example, an LDPC code given below has rank(H 1 H 1 T) = 1 and girth 6:
and requires only one ebit to be shared between source and destination. Since arbitrary classical codes can be used with this approach, including LDPC code of girth g≥6, the performance of quantum LDPC codes can significantly be improved.Because two Pauli operators on n-qubits commute if and only if there is an even number of places in which they differ (neither of which is the identity I operator), we can extend the generators in A (for H 1) by adding e = 1 column so that they can be embedded into a larger Abelian group, the procedure is known as Abelianization in abstract algebra. Someone may use stabilizer version of Gram-Schmidt orthogonalization algorithm to simplify this procedure, as indicated in [13]. We have shown in [12] that decoder for arbitrary quantum check matrix A can be implemented based only on Hadamard and CNOT gates. In Section 2, we have already shown how to implement these two gates in integrated optics. Therefore, arbitrary quantum LDPC encoder and decoder can be implemented in integrated optics.
For example, by performing Gauss-Jordan elimination the quantum check matrix (12) can be put in standard form (see [10] for definition of standard form representation of quantum check matrix):
The corresponding generators in standard form are:where the subscripts are used to denote the positions of corresponding X- and Z-operators. The encoding circuit is shown in Fig. 4(a) , while the decoding circuit is shown in Fig. 4(b). We use the efficient implementation of encoders and decoders introduced by Gottesman [14]. It is clear from Fig. 4 that for encoder and decoder implementation of quantum LDPC codes Hadamard (H) and CNOT (⊕) gates are sufficient, whose implementation in all-fiber technology is already discussed in Section 2.Fig. 4 Encoding and decoding circuits for quantum (6,2) LDPC code: (a) encoder cofiguration, and (b) decoder configuration. |δ1δ2〉 represents two information qubits.
For completeness of the paper, in Fig. 5 we provide comparison of EA LDPC codes of girth g = 6 and 8 against dual-containing LDPC code (g = 4). We see that EA LDPC codes outperform for more than order in magnitude the corresponding dual-containing LDPC code.
5. Conclusions
We have shown that different sets of universal quantum gates can be implemented in all-fiber technology. We have also shown how to implement Pauli operators (needed in QECC) in the same technology. The basic building blocks for implementation of various photonic quantum gates are directional couplers and HNLF. The implementation of the entanglement EPR preparation circuit in all-fiber technology has been described as well. Because the quantum information processing relies on delicate superposition states, it is quite sensitive to quantum errors. We have shown that the encoders and decoders for arbitrary quantum error correcting code can be implemented based on directional couplers and HNLFs. We provide an example to illustrate the encoder/decoder implementation in all-fiber technology (or integrated optics). Finally, we demonstrate by simulation that EA sparse-graph codes significantly outperform dual-containing LDPC codes.
Acknowledgments
This work was supported in part by the National Science Foundation (NSF) under Grant IHCS-0725405.
* The girth represents the shortest cycle in corresponding bipartite graph representation of a parity-check matrix of classical code.
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