The ability to encode information on single photons has changed how information can be processed or transferred fundamentally, enabling advanced technologies such as quantum communication or quantum computing [1–5]. Empowered by this ability, a global network capable of exchanging information between quantum devices is now being envisioned [6]. The scalability of such a quantum network will hence be strictly limited by how much information a photon can carry.

To encode a large quantity of information on a photon, one may harness various degrees of freedom (DOF) of a single photon, including the optical angular momentum, frequency bins, or time bins [7–14]. Alternatively, the information can be encoded in high dimensions [15–21], such as the optical modes of a single physical DOF. It is also proposed that, with the computational space largely expanded by the high-dimensional encoding, powerful information processing can be realized with only “one” photon in high dimensions [22–26]. Nevertheless, precise control over various optical properties in large dimensions is very challenging at the single-photon level. The experimental implementation of quantum algorithms such as Shor’s algorithm using one high-dimensional photon alone has never been achieved.

In this paper, we explore the information encoding and manipulation of a high-dimensional single photon in a single physical DOF. We demonstrate the encoding of information on a temporally long single photon across 32 time bins or dimensions, which is the largest reported to date for a time-bin-encoded single photon. The high-dimensional quantum state is efficiently prepared by modulating the single-photon wave packet in a single shot. By manipulating the high-dimensional single-photon state with a compact interferometry and time-resolved detection, we experimentally realize a compiled version of Shor’s algorithm [27–30] and the factorization of the integer 15 using a single photon for the first time. Shor’s algorithm [31] is the most sophisticated quantum algorithm with an exponential speedup on the factorization problem. Our work thus demonstrates not only the astonishing information capacity of a single photon but also the powerful information processing of a single photon in high dimensions.

The compiled version of Shor’s algorithm [27–30] is designed to find the prime factors of a specific integer 𝑁 or, equivalently, the order 𝑟 of 𝑎modulo𝑁 such that 1≤𝑟<𝑁 and 𝑎𝑟≡1⁢mod⁡𝑁 for any coprime integers 𝑎 and 𝑁. The algorithm consists of three steps. The register initialization prepares the first (argument) register in an equal coherent superposition state,

The modular exponentiation then produces the highly entangled state,

by applying the order-finding function 𝑎𝑥⁢mod⁡𝑁 to the second (function) register when the argument register is in the state |𝑥⟩. Finally, the inverse quantum Fourier transform (QFT) is applied to the argument register,

followed by the measurement in the computational basis to find the interference peak at |𝑦⟩ with 𝑦=𝑐⁢2𝑛/𝑟.

To encode the computational basis states on a single photon, we exploit the time-bin modes 𝑑𝑗 within its temporal wave packet,

where 2𝑛 is the number of dimensions and 𝑐𝑗 is the probability amplitude of the 𝑗⁢th time-bin mode. To manipulate the time-bin modes, the single photon is prepared in an equal superposition state before entering a loop structure of programmable phase shifters and time-bin mode couplers (Fig. 1). The phase shifter implements the phase modulation 𝑒𝑖⁢𝜙 for individual time-bin mode. The time-bin mode coupler, composed of sequential operations of full polarization rotation (pol. switch), time-bin displacement (time delay), and partial polarization rotation, couples different time-bin modes 𝑑𝑛 and 𝑑𝑚 [32] as follows:

where 𝐶 is the coupling strength. Consider a constant time delay such that the mode coupling exists only between the nearest-neighboring time-bin modes. The repeated operations structured in Fig. 1(b), where the blue (red) lines represents the phase shifters (mode couplers), generates arbitrary state in the form of Eq. (4) [32–34]. If more efficient coupling between the “far-away” time-bin modes is needed, tunable time-bin displacement can be exploited as shown in Fig. 1(c).

FIG. 1.

The elementary gates can be realized with the phase shifters and mode couplers. Consider a three-qubit state encoded in eight time-bin modes. The single-qubit unitary operations is realized in Fig. 1(d). Targeted on different qubit, the operation is implemented by the mode coupling with different time-bin displacements. For arbitrary single-qubit gates, which can be decomposed into the single-qubit rotations and , the programmable phase shift and mode-coupling operations in Fig. 1(e) are employed. For general two-qubit control unitary operations, the realization for a three-qubit state encoded in eight time-bin modes is shown in Fig. 1(f). The time-bin displacement is again dependent on the target qubit. As an example, Fig. 1(g) shows how for a two-qubit state encoded in four time-bin modes can be implemented by swapping the states and .

Figure 2 shows the quantum circuit used for our implementation of Shor’s algorithm with and . The state is initialized to

after the register initialization and evolves into

after the modular exponentiation. Since for natural numbers , the inverse QFT can be carried out by the classical processing [28–30,35].

FIG. 2.

The experimental setup for implementing Shor’s algorithm is illustrated in Fig. 3. Type-II phase-matched bipthons are generated by the doubly resonant parametric down-conversion in a monolithic PPKTP crystal [36,37]. By pumping the crystal with a cw laser at 775 nm, the detection of one photon heralds a single photon at 1550 nm. The wave packet of the heralded single photon, measured by the time-resolved coincidence counting and shown in Fig. 5(a), has a coherence time of 148 ns with a bandwidth of 2.7 MHz [38] and pair-generation rate of . The temporally long wave packet not only enables the manipulation of the time-bin modes by electro-optic modulators (2-dB insertion loss) [39–45] but also allows us to initialize the state in Eq. (6) by the electro-optic amplitude modulation in just one shot. This shows the advantage of using narrowband single photons to generate a large number of time bins as compared to multipassing the loop structure [34,46,47] in terms of the setup complexity and scalability. It is also worth noting that, to prepare the initial state in Eq. (6), no phase modulation is needed as all the time bins are in phase.

FIG. 3.

FIG. 4.

To simplify the implementation of the gate operations and minimize the optical loss associated with the fiber-based devices, optical circulator and fiber optic retroreflector are used to enable the multipassing of the optical loop in a more compact architecture. The two cnot gates in the modular exponentiation are implemented by an electro-optic polarization switch (3.5-dB insertion loss) and optical delays (fibers). While the first cnot gate double passes the optical delay by using two fiber-based polarizing beam splitters, the second cnot gate passes the optical delay once. After the modular exponentiation, the outcome is analyzed by a time digitizer (100-ps time resolution) and processed by a computer to reveal the result of the factorization.

The single-qubit and two-qubit gates, which compose an universal gate set [48], are carefully characterized in Fig. 4. The measured single-qubit rotations 𝑒𝑖⁢𝜙⁢ˆ𝜎𝑦 and 𝑒𝑖⁢𝜙⁢ˆ𝜎𝑧 on a Bloch sphere are shown in Figs. 4(a) and 4(b), respectively. These single-qubit rotations are used not only for implementing arbitrary single-qubit gates but also for calibrating the modulation signals. The measured truth table of the CNOT10 gate is shown in Fig. 4(c), which is in good agreement with the theoretical truth table in Fig. 4(d).

The experimental demonstration of Shor’s algorithm with a single photon is summarized in Figs. 5 and 6. As shown in Figs. 5(b) to 5(d), the initial time-bin modes as well as the output-state probabilities after the first cnot gate and modular exponentiation exhibit excellent visibility with uneven amplitudes. The uneven amplitudes are due to the double-exponential single-photon wave packet [Fig. 5(a)] and the different losses experienced by different time-bin modes. Nevertheless, Fig. 6 shows that the outcome of the argument registers 𝑥2⁢𝑥1⁢𝑥0 after the inverse QFT is in good agreement with the theoretical prediction assuming even amplitudes for the initial time-bin modes. This surprising result is a consequence of the Hadamard operations in the inverse QFT, which equally distribute the interference peaks at 000, 010, 100, and 110. These peaks correspond to 𝑥=𝑐⁢23/𝑟=0,2,4,6, respectively, for integer 𝑐. The case 𝑥=0 is an inherent failure of Shor’s algorithm while the case 𝑥=4 gives the trivial result of the factors 1 and 15. The cases 𝑥=2 and 𝑥=6 result in finding the order (periodicity) 𝑟=4 with the factors gcd⁡(𝑎𝑟/2±1,𝑁)=3 and 5.

FIG. 5.

FIG. 6.

In summary, we have implemented Shor’s algorithm with a single photon by encoding and manipulating 32 time-bin modes on the single-photon wave packet. The ability of performing such a complex quantum computing task manifests the powerful information-processing capacity of a single photon in high dimensions. Considering commercially available electro-optic modulators with a 40-GHz bandwidth (EOSPACE, Inc.), encoding more than 5000 time-bin modes on temporally long single photons is possible. The manipulation and noise with the high-dimensional states are usually more troublesome compared to the qubits. Nevertheless, our work shows that the high-dimensional time-bin states of a temporally long single photon can be prepared in a single shot and manipulated by a compact programmable fiber loop. The high-dimensional states may also be manipulated by the high-dimensional quantum gates [8,12] with the single-qudit interferometry replaced by a multiqudit interferometry [34], in which the use of multiple photons provides the scalability. The reduction of the number of single-photon sources and detectors will be beneficial for achieving a higher ratio of the coincidence to accidental counts [21,49–51]. It has also been shown that the high-dimensional states have a higher resistance to quantum channel noise [52–54]. The time-bin-encoded states of temporally long single photons are thus promising for implementing high-dimensional quantum computing.

The authors would like to thank D.-S. Chuu and C.-W. Yang for the insightful discussion. This work was supported by the National Science and Technology Council, Taiwan (110-2112-M-007-021-MY3, 111-2119-M-007-007, and 112-2119-M-007-007).