1. Introduction

Graph states are the essential resource for one-way quantum computation (1WQC) [1, 2]. This involves the preparation of qubits entangled in the shape of a graph followed by sequential measurements on its local qubits. As the computational capacity of a graph depends on size as well as structure, much effort has been made to increase the number of qubits in a graph state. A standard approach using photons is to apply entangling gates between basic two-qubit graphs produced by spontaneous parametric down-conversion (SPDC) [3]. For example, 1WQC with six-qubit graph states has been experimentally demonstrated with a combination of three SPDC photon pair sources [4]. Entanglement of more than six photons has not been feasible because of the limited SPDC efficiency.

Schemes to encode more than one qubit per photon have been developed to further increase the number of qubits given the limitation on the number of photons. Qubits are encoded to multiple degrees of freedom and entangled collectively to generate so-called hybrid entanglement [5]. For example, single-photon two-qubit (1P2Q) and single-photon three-qubit (1P3Q) schemes using both polarization and photonic path qubits have realized six-qubit graph states [6, 7]. However, their scalability is constrained by the experimental difficulties that limit path qubits to dangling nodes connected to polarization qubits at the final step of generating a graph [6, 8, 9] (except in two-photon hyper-entanglement experiments [7, 10–12].)

In this paper, we present an approach that implements a fusion of path qubits from different photons. A new class of entangled states can be generated by fusing separate 1P2Q-based graph states. We demonstrate a fusion of two separate two-photon four-qubit (2P4Q) linear graph states, resulting in genuine seven-qubit entanglement, which is verified by evaluating the entanglement witness. The fused graph state is used to demonstrate a six-qubit 1WQC protocol for the two-bit Deutsch-Jozsa algorithm [13].

2. Method

We generate a seven-qubit graph state composed of polarization qubits and path qubits by employing a path qubit fusion gate to join two 2P4Q graphs. Figure 1(a) shows the conceptual scheme of a path qubit fusion gate. The structure in Fig. 1(a) has the same physical topology as the type-I fusion gate proposed for polarization qubits [3]. Two input photons respectively propagate along two paths that represent |0〉_i and |1〉_i, where i denotes the i-th photon. The paths |1〉₁ and |0〉₂ are superposed by a 50:50 non-polarizing beam splitter (NPBS). Post-selection of the case where only one photon arrives at the single photon counter (SPC) in Fig. 1(a) corresponds to the projection |0〉〈00|₁₂ + |1〉〈11|₁₂ with the initial paths |0〉₁ and |1〉₂ respectively becoming the paths |0〉 and |1〉 of the fused path qubit. When the two photons carry both polarization qubits and path qubits, the two polarization qubits are interchanged only if the fused path qubit is |1〉. In other words, the fusion gate shown in Fig. 1(a) consists of a fusion of the path qubits followed by a controlled swap operation on the polarization qubits. Here, the fused path qubit becomes the control qubit. Therefore, a pure fusion operation requires the fusion gate to be applied to two photons whose polarization qubits are symmetric and unaffected by the swap operation, as described in the next section. The actual implementation of the path qubit fusion gate in our scheme uses a birefringent prism (BP) instead of an NPBS as in Fig. 1(a), and the technical details are explained in Appendix C.

 

Fig. 1 Seven-qubit graph state generation and measurement. (a) Schematic of path qubit fusion gate. (b) Generation of four-photon seven-qubit graph state from two two-photon four-qubit linear graph states. Polarization qubit p_i and path (spatial) qubit s_i are encoded in photon i. FG: fusion gate. (c) Overall experimental setup. BP: birefringent prism, Q: quarter-wave plate, H: half-wave plate, P: polarizer, SMF: single-mode fiber, NPBS: non-polarizing beam splitter, BCQ: birefringence-compensating quartz crystal, PFQ: polarization-flipping quartz crystal, PS: phase shifter, WOC: walkoff compensator, SPC: single-photon counter (PerkinElmer SPCM-AQ4C), IF: interference filter (10 nm for SPC1 and SPC2, 5 nm for SPC3 and SPC4). The angles of the wave plates denote the direction of the slow axis with respect to the horizontal.

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A fusion gate applied to path qubits located in the middle of separate 2P4Q graphs generates a 4P7Q state as depicted in Fig. 1(b). We briefly describe our 2P4Q state generator. (Details can be found in [14]). Initially, two photons are prepared in the Bell state $\left(|HH〉+|VV〉\right)/\sqrt{2}$. The separate paths of the two photons are respectively split by a polarizing beam splitter (PBS) and an NPBS. One output path from the NPBS passes through a half-wave plate (HWP) that interchanges horizontal and vertical polarizations. The resulting 2P4Q state is a four-qubit linear graph state as shown in the left part of Fig. 1(b). In both [14] and our current scheme, the computational basis states |0〉 and |1〉 are respectively encoded onto the linear polarizations along −45° and +45°, and also encoded onto the two output paths of the PBS and the NPBS.

3. Experimental setup and results

Figure 1(c) shows the overall experimental setup, from the generation of separate 2P4Q states to the detection of the 4P7Q graph state. The Bell-state photon pairs are generated by type-I SPDC in cascaded BBO crystals [15, 16]. The pre-walkoff-compensated [16] pump laser (wavelength 390 nm, pulse duration 200 fs, repetition 76 MHz, average power 300 mW) propagates in a double-pass geometry through the crystals and generates two photon pairs, in the forward and backward directions, respectively. The four photons are coincidence-counted by four single photon counters (SPC1∼SPC4). The photon detected by SPCi is henceforth labeled as photon i.

High phase stability for the path qubits is achieved by replacing the PBS-based scheme used in the original 2P4Q state generator [14] with a compact scheme using a birefringent prism (BP1) that serves a twofold function: (i) generation of two 2P4Q graph states through polarization-dependent beam separation, and (ii) path qubit fusion by combining two separate beam paths for distinct photons (see Appendix C). Path combination by BP1 is possible because the respective polarizations in the two paths are mutually orthogonal. The use of a BP instead of an NPBS for path qubit fusion has the benefit of obviating the 50% photon loss associated with the NPBS. After the fusion operation of BP1, the photon sent to SPC4 along the combined path (photon 4) carries only a polarization qubit. The remaining two paths that lead to SPC3 correspond to paths |0〉 and |1〉 for photon 3.

The other two photons, photons 1 and 2, are sent along paths directed to detectors SPC1 and SPC2, respectively. Each photon is spatial-mode-filtered by a single-mode fiber (SMF) before entering an NPBS that splits the incoming path into a Sagnac interferometric configuration. One output path from each NPBS passes through a pair of birefringent quartz crystals (BCQs) whose optic axes are horizontally and vertically aligned, respectively, to compensate for unwanted birefringence from optical components [14]. Polarization flips, required for 2P4Q state generation, are performed by a pair of birefringent quartz crystals (PFQs) whose optic axes are aligned along 45° or −45° from the horizontal axis. The PFQ crystals are tilted to yield a combined birefringence equivalent to an HWP with its slow axis aligned along 45°.

The measurement of the polarization qubits p₁ ∼ p₄ follows standard procedure with the use of HWPs, quarter-wave plates (QWPs), and polarizers. Path qubit measurements, however, are configured differently depending on the photon. The path qubits s₁ and s₂ for photons 1 and 2 are each projected and measured with a Sagnac interferometer whose NPBS initially splits the incoming path and subsequently recombines the interfering paths, as illustrated in the lower part of Fig. 1(c). The path qubit is projected to |0〉 or |1〉 by blocking either path with a shutter (not shown in Fig. 1(c)), or projected to the superposed state $\left(|0〉+e^{i\varphi }|1〉\right)/\sqrt{2}$ by adjusting the phase ϕ with a tiltable 1-mm-thick glass plate (PS). The phase stability for path qubits s₁ and s₂ is maintained by interferometers with Sagnac geometry.

The path qubit s₃ is measured by applying a birefringent prism (BP2) to combine the two paths for photon 3. The path combination is preceded by an interchange of the horizontal and vertical polarizations performed by an HWP before BP2, and followed by another interchange of polarizations through a QWP-HWP-QWP sequence. This double-swap of polarizations has been introduced to match the path lengths for photons 3 and 4: matching the lengths of the two paths for either photon ensures path-length matching for the other photon. The QWP-HWP-QWP sequence is used instead of a single HWP because the HWP between the two QWPs can control the relative phase between the two paths in path qubit measurement. Rotation of this HWP changes the phase ϕ of the superposition state $\left(|0〉+e^{i\varphi }|1〉\right)/\sqrt{2}$ since the polarization of the |0〉 (|1〉) path is fixed as horizontal (vertical). A stable phase is maintained because the interfering paths share most of their optical components. Projection to |0〉 or |1〉 is done by suitably placing a shutter in front of BP2. By using a BP rather than an NPBS for path qubit measurements, we achieve twice as many coincidence counts at the cost of dispensing with the detection of the |V〉 (|H〉) component present in the |0〉 (|1〉) path. We note that BP1 suppresses the amount of the undetected components to below 0.03%, which leads to an error significantly less than the statistical uncertainty of our measurements. A similar path qubit measurement scheme using a PBS has been reported in the literature [9].

The operation of the experimental setup is critically dependent on successful path qubit fusion. The stability of the fused path qubit is tested by measuring the coherence between the |0〉 and |1〉 states of s₃. All the polarization qubits adjacent to path qubit s₃ are projected to |0〉, which ideally projects qubit s₃ to |+〉, and the path qubits s₁ and s₂ are projected to |+〉 to maximize the four-photon count rate. The coherence of s₃ is measured by four-photon coincidence counts with varying relative phase as shown in Fig. 2. The coherence between |0〉 and |1〉 is visibly maintained for a total measurement time of 5 hours (23 data points, 800 s each) with an interference visibility of 0.52 ± 0.03. The phase of the interference fringes drifts by less than 2 × 10°/h during our experiments with a temperature dependence of ∼ 2 × 100°/K (see Appendix E). We therefore re-adjust the phase offset every 4 or 5 hours, which limits the reduction in visibility induced by the phase drift to less than 10%. The mechanism of the temperature dependence is currently under investigation.

 

Fig. 2 Interference between the |0〉- and |1〉-components of fused path qubit. Four-fold coincidence counts in 800 s with qubits p₁ ∼ p₄ and s₁ ∼ s₂ projected to |0〉 and |+〉, respectively, and qubit s₃ projected to $\left(|0〉+e^{i\varphi }|1〉\right)/\sqrt{2}$. Error bars denote ± $\sqrt{\text{counts}}$.

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The non-ideal visibility in Fig. 2 is due to experimental imperfections. Since the interference is generated by two independent photons (photons 3 and 4), the visibility is limited by the degrees of polarization (DOPs) and the spectral purities of the photons. Unpolarized photons or two photons in different spectral modes contribute to a constant background, therefore the upper bound of the visibility is set by the product of the two DOPs and the probability that the two photons are spectrally indistinguishable. The measured DOP of each photon is 0.9 (see Appendix F), whose departure from the ideal value of 1 may be due to insufficient accuracy in walkoff pre-compensation of the SPDC source. The probability of spectral indistinguishability is 0.67 (see Appendix G). Hence the upper bound of the visibility is 0.54 (= 0.9² × 0.67), which is in fair agreement with the measured visibility. This visibility is comparable to the results of similar photonic interference experiments [8, 17]. A higher visibility could be achieved in our experiment by using filters with narrower bandwidths to enhance the spectral purity. However, the lower photon counting rates lead to longer measurement times.

A definite test of the 4P7Q state is the measurement of the entanglement witness defined as [18]

Table 1. Expectation Values of Stabilizer Operators and Entanglement Witness

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4. Deutsch-Jozsa algorithm for two-bit functions

Our genuinely entangled 4P7Q state is applied to a demonstration of a 1WQC algorithm, specifically the Deutsch-Josza algorithm (DJA) for two-bit functions developed for an E-shaped six-qubit graph state [13]. This six-qubit state is prepared by detaching qubit p₄ from the seven-qubit graph by projection to |0〉 as shown in Fig. 3(a). The DJA scheme is implemented according to the procedure shown in Table 2 of [13]. Qubits p₁, p₂, and p₃ constitute an oracle preparing a function and an ancilla, while qubits s₁, s₂, and s₃ reveal the computation result. The DJA is demonstrated for four functions, f({0, 1, 2, 3}) = {0, 0, 0, 0}, {0, 0, 1, 1}, {0, 1, 0, 1}, {0, 1, 1, 0}, which are labeled (i), (iii), (v), (vii), respectively; the other four functions, f({0, 1, 2, 3}) = {1, 1, 1, 1}, {1, 1, 0, 0}, {1, 0, 1, 0}, {1, 0, 0, 1} require exactly the same measurement bases as (i), (iii), (v), (vii), respectively, hence can be omitted without loss of generality. Depending on which function is selected, a set of measurements (Y or Z, X, X or Y, Z) is performed on qubits p₁ and p₂, p₃, s₁ and s₂, s₃, respectively, and feedforward is applied afterwards. The raw data with the measurement bases are listed in Appendix B and the output probability results are shown in Fig. 3(b). The output probability is greater than 90% when compared with the ideal case where |s₃〉 is always |−〉 and |s₁〉|s₂〉 is |0〉|0〉 only for a constant function.

 

Fig. 3 General two-bit Deutsch-Jozsa algorithm on the 4P7Q state. (a) Structure of graph state and logic flow. (b) Measured output probabilities (%) for ancilla qubit (y) and query qubit (x₁ and x₂). (i), (iii), (v), (vii) are results for functions f({0, 1, 2, 3}) = {0, 0, 0, 0}, {0, 0, 1, 1}, {0, 1, 0, 1}, {0, 1, 1, 0}, respectively.

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Table 2. (a) X-measurement for Path qubits and Z-measurement for Polarization Qubits; Coincidence Counts in 500 s; (b) Z-measurement for Path qubits and X-measurement for Polarization Qubits; Coincidence Counts in 2000 s

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Table 4. Slow Axes Orientations of the Wave Plates

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A high-fidelity graph state leads to a high success probability for the relevant one-way quantum computation. It is also possible for a low-fidelity graph state, with certain restrictions, to compute a quantum algorithm with high success probability. In our experiment, the genuine entanglement of all 7 qubits is limited by the weakest qubit s₃, hence the relatively low quality of the fused qubit s₃ (as inferred from the value of S(s₃) in Table 1) diminishes the overall fidelity of the 7-qubit graph state. The degradation in the quality of s₃ is mostly due to phase errors, which, in principle, do not affect the success probabilities of the DJA. This is because the fused qubit s₃ is an ancilla that is always projected to |0〉 or |1〉 (which by feedforward correnspond to |+〉 or |−〉) as a result of p₃ being measured in the X -basis. Therefore, the success probabilities for the Deutsch-Jozsa algorithm (DJA) are high (> 90%) despite a fairly low estimate (> 64%) for the fidelity of our seven-qubit graph.

5. Conclusion

We have demonstrated that fusion of path qubits from distinct photons is a feasible approach to generating larger and more complex graph states. The realization of one-way quantum computation (1WQC) with a seven-qubit graph state is enabled by path qubit fusion. Our results extend the previous work on the two-bit Deutsch-Jozsa algorithm (DJA) with two functions [20] to encompass all functions that are constant or balanced. To our knowledge, this is the first 1WQC demonstration of the DJA for general two-bit functions. Our method of combining path qubit fusion with two-photon four-qubit state generation is applicable to other schemes that use path qubits [8, 9]. We expect this strategy for scaling up graph states to be useful for the generation of graph states with structures suitable for other 1WQC algorithms.

Appendix

A. Raw data for entanglement witness

Table 2 shows the raw data for estimating the entanglement witness of the hybrid seven-qubit graph state.

B. Raw data for two-bit Deutsch-Jozsa Algorithm

Table 3 shows the raw data obtained during the two-bit DJA execution. A constant function f({0, 1, 2, 3}) = {0, 0, 0, 0} is labeled (i). Balanced functions f({0, 1, 2, 3}) = {0, 0, 1, 1}, {0, 1, 0, 1}, {0, 1, 1, 0} are labeled (iii), (v), (vii), respectively.

Table 3. Measurement Bases and Measured Coincidence Counts in 1000 s for DJA Execution

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C. Comparison of a non-polarizing beam splitter with a birefringent prism for path qubit fusion

We compare two implementations for a path qubit fusion gate: a scheme using a non-polarizing beam splitter (NPBS) as in Fig. 1(a), and a scheme using a birefringent prism (BP) as in Fig. 1(c). Let the initial polarization-path state be expressed as

(2)$$\begin{array}{r}\hfill |\psi 〉=a|{\mathrm{\Phi }}_a〉{|0〉}_{s_1}{|0〉}_{s_2}+b|{\mathrm{\Phi }}_b〉{|0〉}_{s_1}{|1〉}_{s_2}\\ \hfill +c|{\mathrm{\Phi }}_c〉{|1〉}_{s_1}{|0〉}_{s_2}+d|{\mathrm{\Phi }}_d〉{|1〉}_{s_1}{|1〉}_{s_2},\end{array}$$

where a, b, c, and d are constants, and |Φ_a,b,c,d〉 denotes the state of the qubits excluding s₁ and s₂. Here, the subscripts i = 1, 2 in s_i are labels for the two photons to be fused. A path qubit fusion gate using an NPBS post-selects the following (unnormalized) state based on two-photon coincidence detection:

(3)$${|\psi 〉}_{\mathit{NPBS}}=\frac{a}{\sqrt{2}}{|\mathrm{\Phi }〉}_a{|0〉}_s+\frac{d}{\sqrt{2}}{𝒰}_{p_1{p}_2}|{\mathrm{\Phi }}_d〉{|1〉}_s,$$

where s, p₁, and p₂ respectively denote the fused path qubit, the polarization qubit of photon 1, and the polarization qubit of photon 2. 𝒰_ij is a swap gate that is defined as 𝒰_ij|α〉_i|β〉_j = |β〉_i|α〉_j for any pure states |α〉 and |β〉. Note that the photon labels in Eq. (3) are different from the labeling used in Fig. 1(c): the photon from an NPBS output port and detected by the single-photon counter (SPC) in Fig. 1(a) corresponds to photon 2, and the other photon in Fig. 1(a) corresponds to photon 1. The transformation from Eq. (2) to Eq. (3) is a path qubit fusion |0〉_s〈0|_s1 〈0|_s2 + |1〉_s〈1|_s1 〈1|_s2 followed by a controlled swap operation on the polarization qubits, with the fused path qubit being the control qubit.

When a BP replaces the NPBS, the post-selection transforms the initial state to

(4)$$\begin{array}{ll}{|\psi 〉}_{BP}\hfill & =a\left({|H〉}_{p_2}{〈H|}_{p_2}|{\mathrm{\Phi }}_a〉\right){|0〉}_s\hfill \\ \hfill & +d{𝒰}_{p_1{p}_2}\left({|V〉}_{p_1}{|V〉}_{p_1}{〈V|}_{p_1}|{\mathrm{\Phi }}_d〉\right){|1〉}_s,\hfill \end{array}$$

which has an additional polarization state projection compared to Eq. (3). In our case, the input polarization state of p₁ (p₂) is fixed as |V〉 (|H〉) in the |1〉_s1 (0〉|_s2) path because the input paths are generated earlier within BP1. Therefore, the polarization state projection does not alter the output state, and Eq. (4) becomes equivalent to Eq. (3) up to a constant factor.

D. Wave plate settings for polarization qubit measurements

In our scheme, the polarization qubits |0〉 and |1〉 are encoded as linear polarizations along −45° and +45°, respectively. The polarizers in front of the fiber-coupling lenses transmit horizontal polarization. The slow axes of the half-wave plate (HWP) and the quarter-wave plate (QWP) are aligned according to the following table for polarization qubit measurements.

E. Stability of path-qubit fusion gate

The phase offset of the interference fringes shown in Fig. 2 are monitored during the witness measurement. As shown in Fig. 4, the phase offset changes according to the ambient temperature which varies with a period of 24 h. The proportionality constant is ∼ 2 × 100°/K. The effect of this phase drift is estimated by recalculating the visibility using only one period, consisting of 12 data points, from the 23-point data in Fig. 2. We find the recalculation to increase the visibility to 0.59 ± 0.07. From this result and the slope of the phase offset graph in Fig. 4, we estimate that the reduction in visibility due to phase instability is less than 10%. The mechanism of this temperature dependence is currently unclear and under investigation. The phase drift is larger than expected since another spatial interferometer using the same type of birefringent prisms has shown a much higher level of stability (∼ 1°/13 h) [21]. We note, however, that resetting the phase offset every 4 or 5 hours without any active stabilization provides sufficient stability for all the measurements in our work.

 

Fig. 4 Phase offset of the interference fringes from the fused path qubit with concurrent ambient temperature.

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F. Degree of polarization of photons

The degree of polarization of the two input photons of the path qubit fusion gate is closely related to the quality of the initial Bell states $1/\sqrt{2}\left(|HH〉+|VV〉\right)$ that charaterizes the photon pairs are emitted in the forward and backward directions. Since the photon pairs are produced by type-I SPDC in double-crystal geometry, the major degradation in quality arises by decoherence between |HH〉 and |VV〉 when the polarization-dependent walkoff in SPDC crystals is not perfectly compensated. Therefore the degree of polarization of photon 3 or 4 heralded by the detection of 45°-polarized photon 1 or 2, respectively, is obtained as the interference visibility between the |H〉 and |V〉 components. This visibility is approximately the same as the concurrence of the Bell state, and the experimentally measured value is 0.9 for both photons 3 and 4.

G. Spectral purity of photons

The path lengths for the two photons emitted along the upper direction in Fig. 1 are matched by varying the path length of the upper-right photon to measure two-photon bunching at SPC3 and SPC4. The fiber output port to each SPC is connected to a 50:50 fiber directional coupler leading to two photon counters that measure two-photon coincidences, as shown in Fig. 5. The polarization for each port is projected to 45° to equally detect backward- and forward-emitted photons from the SPDC source. Here, photons 1 and 2 in the lower part of Fig. 1 are not measured in order to shorten the measurement duration.

 

Fig. 5 Photon bunching measurement at output ports for (a) SPC3 port and (b) SPC4.

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The spectral purity of the two interfering photons is estimated from the visibility. If the photons are in the same spectral/temporal mode, the two-photon coincidences should increase with path length matching and show a peak higher by 1/3 compared to the background level. As shown in Fig. 5, the measured peak is higher by 28% and 20% than the background at SPC3 and SPC4, respectively. The normalized visibilities, 84% and 60%, correspond to the values of spectral indistinguishability Tr(ρ_fρ_b) between the two photons, respectively, where ρ_f(b) is the spectral-mode density matrix of the forward- (backward-) emitted photon in the upper part of Fig. 1. The visibility approximately equals the spectral purity Tr(ρ²) for each photon. Since Fig. 5 is obtained using unheralded photons, we calculate the purity of heralded photons by considering the pump laser configuration and the filter bandwidth (10 nm) of heralding photons [22]. The calculation results are 0.91 and 0.74 at SPC3 and SPC4, respectively. As each purity denotes the probability that the two incoming wave packets at each port are indistinguishable, the product 0.67 of the two purities becomes the probability that the two interfering bi-photon states are spectrally indistinguishable, i.e., coherent.

Acknowledgments

This work has been supported by the KRISS project ‘Single-Quantum-Based Metrology in Nanoscale.’

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