Pairwise fusion gate formed using linear optical circuit

Abstract

The present disclosure provides a mounting method in which is used a linear optical circuit for quantum measurement that is a so-called pairwise fusion gate (PFG) having a function for stochastically projecting two inputted qudits onto the quantum entangled state of a two-qubit subspace. Specifically, provided is a pairwise fusion gate characterized by comprising a linear optical element and a photon detector, the pairwise fusion gate furthermore being characterized in that: the linear optical element and the photon detector are configured so that an input which has two d-dimensional qudits defined using d-rail coding (where d is an integer equal to or greater than 3), and in which there is no quantum entanglement between the two qudits, is stochastically projected onto a quantum entanglement state in a two-qubit subspace; and measurement results from the photon detector are used in determining the specific two-qubit subspace in which the projection onto the quantum entanglement state has been made.

Description

Pairwise Fusion Gate by Linear Optical Circuit

[0001] This disclosure relates to a method for realizing quantum operations on photonic qubits using linear optical elements and photon detectors.

[0002] Quantum information processing using photons can be performed by defining |0> and |1> of a qubit according to whether a single photon exists in either of two modes. The modes used here include polarization, time bin, frequency bin, orbital angular momentum, frequency comb-like modes, and those using spatial modes. For example, when using polarization, a qubit is formed depending on whether a single photon is horizontally polarized or vertically polarized. When using a time bin, a qubit can be formed using two temporally separated single photon beams.

[0003] A qubit (photonic qubit) using a single photon can be subjected to quantum operations by using linear optical elements and photon detectors. Here, a linear optical element is an element such as a beam splitter or a phase shifter whose action can be expressed as a linear transformation of a photon creation operator. A photon detector is a device capable of discriminating the number of photons. However, it also includes those capable of discriminating whether the number of photons is 0 or 1 or more.

[0004] In particular, for quantum measurement on two qubits, those that project onto a Bell state with an average success probability of 50% can be realized by a linear optical circuit using linear optical elements and photon detectors. (See Non-Patent Documents 1 and 2). Furthermore, it is possible to improve this success probability by additionally using pre-prepared photons. For example, by using 2(2 k −1) photons, an average success probability of 1−2 -(k+1) can be achieved. (See Non-Patent Document 3).

[0005] By using d modes of a single photon, it is possible to construct a quantum didt (a d-dimensional quantum system) instead of a qubit. For example, in the case of a time-bin quantum didt, |i> is the state in which a photon exists in the i-th wave packet among the d wave packets. Here, i = 0, ..., d-1. Even in cases other than the time bin (except when the number of possible modes such as polarization is limited), it is possible to easily extend photon qubits to photon quantum didts.

[0006]

[0007] Photon quantum dits can also be used in quantum information processing based on qubits. A method called super-entanglement-assisted Bell state analysis (see Non-Patent Literature 7 and 8) uses a linear optical circuit to deterministically identify the Bell state on a two-qubit subsystem of two input quantum quats (i.e., quantum dits with d=4). However, this method requires that the input quantum quats are quantum entangled in the other two-qubit subsystem, and therefore cannot be used for large-scale quantum information processing.

[0008] H. Weinfurter, Experimental bell-state analysis, EPL 25, 559 (1994).S. L. Braunstein and A. Mann, Measurement of the Bell operator and quantum teleportation, Phys. Rev. A 51, R1727 (1995).W. P. Grice, Arbitrarily complete Bell-state measurementusing only linear optical elements, Phys. Rev. A 84, 042331 (2011).Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, A. Zeilinger, and J.-W. Pan, Quantum Teleportation in High Dimensions, Phys. Rev. Lett. 123, 070505 (2019).H. Zhang, C. Zhang, X.-M. Hu, B.-H. Liu, Y.-F. Huang, C.-F. Li, and G.-C. Guo, Arbitrary two-particle high- dimensional Bell-state measurement by auxiliary entanglement, Phys. Rev. A 99, 052301 (2019).N. Bharos, L. Markovich, and J. Borregaard, Efficient High-Dimensional Entangled State Analyzer with Linear Optics (2024), arxiv:2401.15066 [physics, physics:quantph].S. P. Walborn, S. Padua, and C. H. Monken, Hyperentanglement-assisted Bell-state analysis, Phys. Rev.A 68, 042313 (2003).C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, Complete Deterministic Linear Optics Bell State Analysis, Phys. Rev. Lett. 96, 190501 (2006).Tomohiro Yamazaki1 and Koji Azuma, Linear-optical fusion boosted by high-dimensional entanglement, arXiv:2407.10893 (2024).

[0009] This disclosure provides a configuration using a linear optical circuit for quantum measurement called a pairwise fusion gate (PFG), which probabilistically projects two d-dimensional quantum dits onto a Bell state on a two-qubit subspace. Specific implementation methods are described in 1-d -1 We provide a linear-optical circuit (LOC) that operates as a PFG with a success probability of . In another embodiment, for any positive integer k, 2(2 k Success probability using -1) auxiliary photons: 1-d -(k+1) A linear optical circuit that functions as a PFG is provided.

[0010] By applying a linear optical circuit to one photon of each of the two d-dimensional Bell states, 1-d -1 With a certain probability, the state of the two unmeasured photons becomes a two-dimensional Bell state. When d=2, it reduces to a typical setup using a photon qubit, with a success rate of 50%. Therefore, the linear optical circuit of this disclosure functions as an entangled quantum measurement with an improved success rate by using a quantum dut with high-dimensional quantum entanglement as the input state. This is a novel application of photon quantum duts that can also be applied to large-scale quantum information processing.

[0011] This figure shows a graph representation of a linear optical circuit that realizes a pairwise fusion gate (PFG) in one embodiment. This figure shows a graph representation of a linear optical circuit that realizes a PFG in another embodiment when k=1. A d-dimensional Bell state consisting of two photons is used as an auxiliary photon. This figure shows a graph representation of a linear optical circuit that realizes a PFG in another embodiment when k=2. A d-dimensional Bell state consisting of two photons and a d-dimensional 4-body GHZ state consisting of four photons are used as auxiliary photons. This figure shows the experimental setup when realizing the linear optical circuit that realizes a PFG in one embodiment using a time-bin quantum dit. This figure shows the experimental setup when realizing the case where a d-dimensional Bell state consisting of two photons (k=1) is used as an auxiliary photon in the linear optical circuit that realizes a PFG in another embodiment using a time-bin quantum dit.

[0012]

[0013]

[0014] To perform quantum information processing, we define qubits using photons. Here, we use an encoding called "d-rail encoding," defined in Equation 1 below. Here, i = 0, ..., d-1, the left side represents the state of a quantum dit, and the right side represents the state of a single photon.

[0015]

[0016] The linear optical circuit proposed in this disclosure takes two photons of states corresponding to two quantum dits as input, and ultimately measures all modes by a photon detector, returning the measurement pattern of each photon detector as an output. Such a linear optical circuit constitutes a quantum measurement on the input two quantum dits.

[0017] Before describing specific embodiments, the graphical representation of the linear optical circuit used in this disclosure will be explained. (In the graphical representations shown in Figures 1 to 3) Photon modes are drawn with horizontal lines, corresponding to the 0th mode, the 1st mode, and so on, from top to bottom. Photon detectors are represented by deformed semicircles. Interference zones of two-mode linear optical circuits, where the transfer matrix is ​​represented by the Hadamard matrix H, are represented by two connected black dots. Furthermore, the white circles at the left end of the horizontal lines are connected to indicate which modes constitute each quantum dit. In particular, when the input state is fixed, the state of the corresponding quantum dit is described. The graphical representation in this disclosure describes the case where d=3, but the linear optical circuit itself can be easily extended to the case of any dimension d.

[0018] (Embodiment 1)

[0019]

[0020]

[0021] (Embodiment 2)

[0022]

[0023]

[0024]

[0025]

[0026]

[0027]

[0028] As shown in Figure 4, according to one embodiment, the PFG 100a includes a beam splitter 103 having a time-independent 50:50 splitting ratio between two path modes 101a and 101b, and two time-resolved photon detectors 105a and 105b for detecting photons. Two photons corresponding to two input quantum dits enter a superposition state between path modes 101a and 101b in the beam splitter 103, and are then detected by the two detectors 105a and 105b. Although not shown, the optical output detected by the detectors is analyzed and evaluated by a computer or the like.

[0029] When the two photon detectors 105a and 105b detect a total of two photons in the i-th time bin (in the case of (i) of Embodiment 1, Ni = 2), the beam splitter 103 does not change the timing of the photons, which indicates that the two photons input via their respective pass modes 101a and 101b were incident as the i-th time bin at the same timing. Therefore, the beam splitter 103 and the photon detectors 105a and 105b project the two input quantum dits onto the quantum state |i〉|i〉. On the other hand, if either the photon detector 105a or 105b detects one photon each in the i-th time bin and the j-th time bin (in the case of (ii) of Embodiment 1, Ni = 1, Nj = 1), the photon input by the beam splitter 103 is in a superposition state between pass modes 101a and 101b. Therefore, there are two possibilities: one where the photon input via pass mode 101a is incident as the i-th time bin and the photon input via pass mode 101b is incident as the j-th time bin, and another where the photon input via pass mode 101a is incident as the j-th time bin and the photon input via pass mode 101b is incident as the i-th time bin. It is possible that the input two quantum dits are projected onto the superposition state of these two cases. In fact, (according to Non-Patent Literature 9) the state to which the input two quantum dit is projected is determined by the total number of photons detected by the photon detector 105b, regardless of the timing of photon detection. If the number is even, i.e., 0 or 2, it is projected to |Ψi,j,+〉, and if the number is odd, i.e., 1, it is projected to |Ψi,j,-〉.

[0030] Therefore, it becomes possible to probabilistically project a d-dimensional two-qubit dit onto a Bell state in a two-qubit subspace. Assuming the timing of the input photons is completely random, the probability that two photons are input at the same time is 1 / d, so the average probability of projection onto a Bell state in a two-qubit subspace is 1 - 1 / d.

[0031] Another embodiment (when k = 1) using auxiliary photons to further improve the success probability is shown in FIG. 5. As shown in FIG. 5, according to another embodiment, the PFG 100b includes four beam splitters 103a to d having a time-independent 50:50 branching ratio between two path modes 101a and 101b into which two quantum dits are input and two path modes 101c and 101d into which a d-dimensional Bell state 106 composed of two auxiliary photons (also called ancilla photons) is input, and four time-resolved photon detectors 105a to b for detecting photons. In the d-dimensional Bell state, two photons respectively input from the path modes 101c and 101d are in a uniform superposition state from i = 0 to d - 1 where both are in the i-th time bin. The total four input photons are in a superposition state by the beam splitters 103a to d and are detected by the four photon detectors 105a to d. Although not shown, the optical output detected by the detector is analyzed and evaluated by an electronic computer or the like.

[0032]

[0033]

[0034]

[0035] Thus, it becomes possible to probabilistically project a d-dimensional two-quantum dit onto a Bell state in a two-qubit subspace. Assuming that the timing of the input photons is completely random, the probability that two photons corresponding to each quantum dit and two auxiliary photons always input at the same timing are all input at the same timing is 1 / d 2 Therefore, the average probability of being projected onto a Bell state in a two-qubit subspace is 1 - 1 / d 2 Since this is a higher value than the probability when no auxiliary photons are used, it can be seen that a PFG with an improved success probability has been realized by using auxiliary photons.

[0036] Similarly, in general cases other than k=1 in Embodiment 2, by utilizing a time-bin quantum dut, it is possible to achieve a PFG with a higher success rate by using a beam splitter with a time-independent 50:50 branching ratio, a time-resolved photon detector, and auxiliary photons in the GHZ state. Furthermore, even when using a photon quantum dut with d-rail coding other than a time-bin quantum dut, it is possible to achieve a PFG by using an appropriate experimental apparatus.

[0037] Furthermore, the contents of the present invention are not limited to photons, but can also be implemented using other bosons (e.g., phonons and magnons). For this reason, the d-rail coding used in the present invention is generally defined using a single boson on d modes, the linear optical element is a device whose operation can be expressed as a linear transformation of the boson generation operator, and the photon detector is a device capable of identifying some or all of the number of boson particles.

[0038] (Additional Considerations) The above description of embodiments of the present invention is presented for illustrative purposes only and is not intended to be exhaustive or to limit to the exact forms disclosed. Those skilled in the art will understand that many modifications and changes are possible in light of the above disclosure.

[0039] Finally, the language used herein has been selected primarily for readability and instructional purposes, and may not be selected to describe or limit the subject matter of the invention. Therefore, the scope of the invention is intended to be limited by the appended claims, not by this detailed description. Accordingly, the disclosure of embodiments of the invention is intended to illustrate, not limit, the scope of the invention as described in the claims.

[0040] 100a, 100b Pairwise fusion gate 101a-d Path modes 103 Beam splitter 105a-d Photon detector 106 d-dimensional Bell state consisting of two auxiliary photons 107 d-dimensional quad-body GHZ state consisting of four auxiliary photons

Claims

1. A pairwise fusion gate comprising a linear optical element and a photon detector, wherein the linear optical element and the photon detector are configured to probabilistically project onto a quantum entanglement state in a two-qubit subspace for an input such that there are two d-dimensional quantum duts defined using d-rail coding (where d is an integer of 3 or more) and there is no quantum entanglement between the two quantum duts, and the measurement result of the photon detector is used to determine which quantum entanglement state in a two-qubit subspace has been projected onto.

2. The pairwise fusion gate according to claim 1, wherein, in order to improve the probability that the two quantum dits are projected onto any of the quantum entangled states in the two-qubit subspace, a specific quantum state is input to the linear optical element and the photon detector in addition to the two quantum dits.

3. The pairwise fusion gate according to claim 1, wherein a beam splitter having a 50:50 branching ratio is used as the linear optical element, and a pairwise Bell state is used as the quantum entanglement state on the two-qubit subspace.

4. The pairwise fusion gate according to claim 2, wherein the linear optical element is a beam splitter having a 50:50 branching ratio, the pairwise Bell state is used as the quantum entanglement state on the two-qubit subspace, and the d-dimensional Bell state or d-dimensional GHZ state on a quantum bit defined using d-rail coding is used as the auxiliary specific quantum state.

Images