Maximal success probabilities of linear-optical quantum gates

Abstract

Numerical optimization is used to design linear-optical devices that implement a desired quantum gate with perfect fidelity, while maximizing the success rate. For the 2-qubit CS (or CNOT) gate, we provide numerical evidence that the maximum success rate is S = 2/27 using two unentangled ancilla resources; interestingly, additional ancilla resources do not increase the success rate. For the 3-qubit Toffoli gate, we show that perfect fidelity is obtained with only three unentangled ancilla photons -less than in any existing scheme -with a maximum S = 0.00340. This compares well with S = (2/27) 2 /2 ≈ 0.00274, obtainable by combining two CNOT gates and a passive quantum filter [1]. The general optimization approach can easily be applied to other areas of interest, such as quantum error correction, cryptography, and metrology [2,3].PACS numbers: 03.67.Lx, 42.50.Dv Linear optics is considered as a viable method for scalable quantum information processing, due in large part to the seminal work of Knill, Laflamme, and Milburn (KLM) [4]. These authors showed that an elementary quantum logic gate on qubits, encoded in photonic states, can be constructed using a combination of linearoptical elements and quantum measurement. The tradeoff in this measurement-assisted scheme is that the gate is properly implemented only when the measurement yields a positive outcome, i.e., the gate is non-deterministic. Soon after the KLM scheme became a paradigm for linear optical quantum computation (LOQC), it became clear that there is a general unresolved theoretical problem of finding the optimal implementation for a desired quantum transformation [5].For the nonlinear sign (NS) gate, which acts on photons in a single optical mode, α 0 |0 + α 1 |1 + α 2 |2 → α 0 |0 + α 1 |1 − α 2 |2 , the maximum success probability without feed-forward has been theoretically proved to be 1/4 [6]. Here we focus on more complicated gates, taking as examples the two-qubit controlled sign (CS) gate (equivalently, the CNOT gate), and the three-qubit Toffoli gate. For these physically important gates, existing theoretical results are limited to upper or lower bounds on the success probability [1,7,8].A linear-optical quantum gate, or state generator (LO-QSG) [5], can be viewed formally as a device implementing a contraction transformation (for ideal detectors) that converts pure input states into desired pure output states. The goal of the optimization problem is to find a proper linear optical network (see Fig. 1), characterized by a unitary matrix U, that performs the desired transformation [9,10]. The problem is naturally partitioned into two tasks: i) finding a subspace of perfect fidelity within the space of all unitary matrices U, and ii) maximizing the success probability within this subspace. While in this paper we address transformations implemented by linear optics, the method is universal and with minor modifications can be successfully applied to any quantum-information problem involving unitary operations combined with measurements. Origina...

Optical hybrid approaches to quantum information

“…In subsequent works, this efficiency was slightly improved [48]. There are also various, more general, treatments of these non-deterministic linear-optics gates deriving bounds on their efficiencies [49][50][51]. Experimental demonstrations were reported as well [52], even entirely in an optical fiber [53].…”

Electro-Optic Frequency Beam Splitters and Tritters for High-Fidelity Photonic Quantum Information Processing

“…metrics, where V d×d denotes the infinite-dimensional unitary V truncated to the d modes of U d×d [35]. Experimentally, the success probability is further degraded by photon loss, an effect absent in an ideal unitary.…”

“…A simple argument suggests that this undesired "scatter probability" is at least (d − 1)/(2d − 1) for a uniform d-mode mixer based on a single EOM (Appendix A). Yet this limitation can be circumvented by considering two EOMs with a pulse shaper sandwiched between them; the spectral phase imparted by the middle stage ensures that the sidebands populated after the first EOM are returned to the computational space after the second one, thereby making it possible to realize a fully deterministic frequency beamsplitter [12].Quantitatively, the performance of a generic frequency multiport V can be compared to the desiredmetrics, where V d×d denotes the infinite-dimensional unitary V truncated to the d modes of U d×d [35]. Experimentally, the success probability is further degraded by photon loss, an effect absent in an ideal unitary.…”

Imperfect linear-optical photonic gates with number-resolving photodetection

“…However, an arbitrary (N c + N a ) × (N c + N a ) complex matrix U of unit spectral norm ( U = 1, where U is the largest singular value of U ) may be shown via the unitary dilation technique [13,19] to be equivalent to an (N c + N a + N v ) × (N c + N a + N v ) unitary matrix having the same success and fidelity, where N v is the number of vacuum modes in the input and output states. Thus, it is very convenient in practice to relax the unitarity condition and consider general matrices U of norm 1, with the understanding that the number of singular values in U different from unity corresponds to the number of vacuum modes that are required to implement that solution [13]. Furthermore, we may consider arbitrary complex matrices U with the rescaling U → U/ U to ensure unit norm.…”

“…We note that M c = 2 computational photons in N c = 4 computational modes are required for any two-qubit gate in the dual-rail representation. Additionally, the minimum ancillary resource required to implement CNOT with perfect fidelity consists of two single-photon ancillas; that is, the CNOT gate requires M a = 2 ancilla photons in N a = 2 ancilla modes [13]. Therefore, we have a total of four photons in six modes in the initial state, and the linear-optical transformation U is a 6 × 6 matrix.…”