Patent ID: 12216381

DESCRIPTION

As discussed above, amongst the myriad quantum systems suitable for information processing, photons have the critical advantage of extremely low decoherence, with minimal interaction with their surrounding environment, and therefore are ideal for quantum communications and networking. This isolation, however, has the downside of also making photon-photon interactions for two-qubit gates difficult and, with linear optics, inherently probabilistic.

In this disclosure, the high dimensionality in two particular DoFs of a single photon—namely, time and frequency, which are both compatible with fiber optical transmission—is used to encode one qudit in each DoF.FIG.2shows an example of a method of generating at least one photon with encoded quantum information in one of more degrees of freedom with multiple dimensions. In step105, a first photon encoded with quantum information in two or more frequency bins and at least one time bin. In step110, a frequency dependent time operation is performed to generate nonseparability between the frequency bins and at least two time bins in the photon. Nonseparability is the same as entanglement, but is used when entanglement is between quantum information encoded in different degrees of freedom in a single photon.

In one example, step105can start by generating a photon with two frequency bins and one time bin, having states expressed as:

❘"\[LeftBracketingBar]"ψ〉=12⁢(❘"\[LeftBracketingBar]"0〉f+❘"\[LeftBracketingBar]"1〉f)⊗❘"\[LeftBracketingBar]"0〉t,where the frequency can be in a superposition of bins 0 or 1, and time is in bin 0. After the frequency dependent delay of step110, in the frequency bin 1, the time bin is shifted to 1 as well, resulting in states of the form:

❘"\[LeftBracketingBar]"ψ〉=12⁢(❘"\[LeftBracketingBar]"0〉f⁢❘"\[LeftBracketingBar]"0〉t+❘"\[LeftBracketingBar]"1〉f⁢❘"\[LeftBracketingBar]"1〉t),which is a nonseparable (entangled) state. After the frequency dependent delay, at least two time bins exist.

Step110may be carried out by a novel quantum SUM gate discussed herebelow.FIG.3shows a diagram of a deterministic quantum operation on an exemplary single photon with encoded high-dimensional quantum information in both time and frequency DoFs, generated by the process ofFIG.2. In particular, the first photon120may be generated by the process ofFIG.2in this example to have two qudits encoded in d time bins and d frequency bins, shown in the graph125. The photon120can be encoded in an arbitrary superposition of different time and frequency bins. The unused time-frequency slots are shown with dashed circles. The first photon120then goes through a deterministic quantum process130, for example, a logic gate, to modify the encodings in the photon. After the deterministic quantum process130operates on the two-qudit state, the orientation of the time-frequency superpositions change to a new two-qudit state, as illustrated in the graph135.

A related process is at work in the advanced optical modulation formats gaining adoption in modern digital communications, where many bits are encoded in a single symbol via modulation of canonically conjugate quadratures, as described in Marin-Palomo, P. et al. “Microresonator-based solitons for massively parallel coherent optical communications.”Nature546, 274-279 (2017).

Since single photons can be generated in a superposition of many time and frequency bins, multiple qubits can be encoded in each DoF, making such photons a favorable platform for deterministic optical quantum information processing on Hilbert spaces dramatically larger than previously demonstrated deterministic qubit-based gates. Due to the limitation of the number of high-dimensional degrees of freedom in photons, the Hilbert space for deterministic operations cannot get exponentially scaled by adding qudits to the system. Instead, the extension of the Hilbert space is carried out by increasing the dimensions in each degree of freedom.

While enabling only linear scaling of the Hilbert space with the number of modes in photons, qudit encoding promises significant potential in the current generation of quantum circuits. It has been shown, for example, in Roa, L., Delgado, A. & Fuentes-Guridi, I. “Optimal conclusive teleportation of quantum states.”Phys. Rev. A68, 022310 (2003), that two-qudit optical gates are useful in transmitting quantum states with higher information content per photon by means of qudit teleportation. Such qudit teleportation requires two-qudit gates which can operate on the different degrees of freedom of a single photon. The photons generated by the embodiments disclosed herein have such functionality.

FIG.4ashows a schematic block diagram of an experimental implementation of the X gate205, which is a building block to two-qudit gates which can operate photons to carry out the method ofFIG.2. In particular, a building block to such functionality is enabling the realization of all single-qudit unitaries. To this end, it is sufficient to demonstrate the generalized Pauli gates X (cyclic shift) and Z (state-dependent phase), which are universal for single-qudit operations, and from which all d-dimensional Weyl operators can be constructed. The Z gate applies a unique phase shift to each of the d basis states, which can be easily executed with a phase modulator and a pulse shaper in the time domain and frequency domain, respectively. Specifically, for the basis state |n(n=0, . . . , d−1), we have Z|n=exp(2πin/d)|n.

Illustrated inFIG.4a, however, is a more challenging X gate, which realizes the transformation X|n=|n⊕1, where ⊕ denotes addition modulo d. Referring toFIG.4a, the functionality of the X gate or cyclic shift gate205is shown in an experimental set-up200to demonstrate its efficacy. The X gate205performs a time-bin operation, which is a quantum operation that acts on the quantum information encoded in the time degree of freedom, regardless of the state in the frequency domain.

The experimental set-up200includes a laser210, a state preparation unit215, the X gate205, and a state projection unit240, and a single photon detector245. The X gate205in this embodiment includes a Mach-Zender modulator switch230, a delay element (interferometer or other known interferometric structure)233and a piezo-electric phase shifter235. In general, the X gate205operates on time bins in three dimensions, a process which corresponds to state-dependent delay. Because the X gate205operates on each photon individually, we can fully characterize its performance with coherent states, and the statistics of the input field have no impact on the principle of operation.

Accordingly, a continuous-wave (CW) laser210cooperates with a state preparation unit215to generate a desired weak coherent state. The state preparation unit215in this embodiment includes an intensity modulator220and a phase modulator225. Specifically, the laser210generates a continuous-wave single-frequency beam. The intensity modulator220carves out three time bins {|0t, |1t, |2t} to generate light pulses. The phase modulator225manipulates their relative phases to complete the preparation of the desired weak coherent state. The time bins are 3 ns wide with Δt=6 ns center-to-center spacing.

To perform the X gate operation, the time bins |0tand |1tare separated from |2tusing a Mach-Zehnder modulator (MZM) switch236and the path232afor time bins |0tand |1tis delayed by 3 bins (18 ns). The interferometer233carries on the delay on the first path232a, and the piezo-electrical phase shifter (PZT) stabilizes the relative phase between the two arms of the interferometer232aand232b. The circle-shaped fibers of interferometer233indicate the delay, with each circle is equivalent to one time-bin delay (6 ns). While most MZM designs are one-port devices, with one of the two output paths terminated, the MZM switch236is a 1×2 version that permits access to both interferometer outputs. Accordingly, it is in principle lossless—as required for a unitary operation. In practice, however, insertion loss reduces throughput, but this is of a technical nature and not fundamental to the method. It will be appreciated that the interferometer233may be replaced by other known interferometric structures.

After the interferometer233performs the path-dependent delay, another 1×2 MZM operated in reverse, (not shown inFIG.4a) can be used to recombine the time bins deterministically. However, in the proof-of-principle experiment ofFIG.4, a 2×2 fiber coupler250is used for recombination, which introduces an additional 3 dB power penalty. For the measurement scheme, the single photon detector245and time interval analyzer255are used with the generated time bins. Further detail regarding the specific exemplary experiment is provided below in the Methods section of this disclosure.

The transformation matrix performed by the X gate205when probed by single time bins yields a computational basis fidelityCof 0.996±0.001, shown inFIG.4b. As such computational-basis-only measurements do not reflect the phase coherence of the operation, the experimental set up uses the state measurement unit240to prepare superposition states as input and interfere the transformed time bins after the X gate205. To this end, the state measurement unit comprises a cascade of 1-bin and 2-bin delay unbalanced interferometers260,265, respectively, operably coupled to the output of the X gate205. In order to combat environmentally induced phase fluctuations in the interferometers260,265and the X gate205, a CW laser is sent in the backwards direction using a feedback phase control loop with PZTs235,260a, and265a. This is discussed further below in the Methods section.

A phase of 0, ϕ and 2ϕ to the time-bins |0t, and |1tand |2tis applied, respectively, with the phase modulator225in the state preparation stage and ϕ is swept from 0 to 2π, obtaining the interference pattern shown inFIG.4c. After subtraction of the background, a visibility of 0.94±0.01 from the maximum and minimum points was calculated, showing strong phase coherence (the ability to preserve and utilize coherent superpositions) between the time bins after the gate. If assuming a channel model consisting of pure depolarizing (white) noise, this visibility can be used to estimate the process fidelityP, findingP=0.92±0.01 for the X gate. Given the ability to perform arbitrary one-qudit operations using combinations of X and Z gates, it follows that it is in principle possible to generate and measure photons in all mutually unbiased bases—an essential capability for high-dimensional quantum key distribution (QKD), which has been proven to offer greater robustness to noise compared to qubit-based QKD and can enable significantly higher secret key rates over metropolitan-scale distances.

This high-performance time-bin X gate205can thus be incorporated into a frequency network to realize deterministic two-qudit gates, where the frequency DoF acts as the control and the time DoF is the target qudit.FIG.5a, for example shows an experimental set up300for two qudit gates for a single photon. For this demonstration, instead of a weak coherent state as with the X gate experimental set up ofFIG.4a, true single photons are used, heralded by detecting the partner photon of a frequency-bin entangled pair generated through spontaneous four-wave mixing in an on-chip silicon nitride microresonator320.

In particular, first and second photons are provided by a photon source305that includes a continuous-wave laser source310, an intensity modulator315and a silicon nitride microring resonator320. The continuous-wave laser source310is the input to an intensity modulator315and cooperates with the intensity modulator315to generate time-bin qudits. The time bins, defined by intensity modulation of the pump via the intensity modulator315, couple into the silicon nitride microring resonator320with a free spectral range (FSR) Δf=380 GHz and resonance linewidths of δf≃250 MHz, generating a biphoton frequency comb. The time-bin and frequency-bin entanglement of such photons have been proven, as discussed, for example, in Jaramillo-Villegas, J. A. et al. “Persistent energy-time entanglement covering multiple resonances of an onchip biphoton frequency comb.”Optica4, 655-658 (2016), Reimer, C. et al. “Generation of multi-photon entangled states with integrated optical frequency comb sources.”Science(80-.). 351, 2-3 (2016), Kues, M. et al. “On-chip generation of high-dimensional entangled quantum states and their coherent control.”Nature546, 622-626 (2017), and

Imany, P. et al. 50-GHz-spaced comb of high-dimensional frequency-bin entangled photons from an on-chip silicon nitride microresonator.Opt. Express26, 1825-1840 (2018), all of which are incorporated herein by reference.

After the silicon nitride microring resonator320, the experimental setup300further includes a pulse shaper325, a heralding bypass335, two single photon detectors340a,340b, and an event timer345. The pulse shaper325is operably coupled to the output of the resonator320. The bypass335is operably coupled between the pulse shaper325and the second photon detector340b. The quantum logic gate330is operably coupled between the pulse shaper325and the first photon detector340a, in parallel to the heralding bypass335. The single photon detectors340a,340bare operably coupled to the event timer345.

The quantum logic gate330is configured to perform a frequency dependent delay to generate non-separable states between the frequency bins and time bins in the photon. In this embodiment, the quantum logic gate330may suitably be a controlled increment (“CINC”) gate330a, or a SUM gate330b. The quantum logic gate330in this embodiment is configured to operation on a photon with three frequency bins |0f, |1f, and |2f.

The CINC gate330aincludes an optical switch352, a first path354a, and one or more other paths. In this example, the optical switch is a dense wavelength division multiplexer (DWDM)352configured to separate frequency bin |2ffrom |0fand |1f. The first path354areceives for frequency bin |2f, and the one or more other paths354breceive frequency bins |0fand |1f. The first path354aincludes an X gate355, configured to introduce a time-bin cyclic shift, and is coupled to an optical frequency combiner360(another DWDM). The X gate355may suitably have the same structure and operation as the X gate ofFIG.4a, namely, having a MZM356, an interferometer357, and a PZT350. The bypass path(s)354bis/are coupled directly to the optical combiner360. The output of the combiner360is operably coupled to the single photon detector340avia another DWDM362.

As discussed above, the elements of the photon generate unit305cooperate to generate first and second photons, referred to herein as signal and idler photons, which are time-bin and frequency-bin entangled. As time- and frequency-bins exceed the Fourier limit (ΔfΔt=2280, δfΔt=1.5), the time-frequency entangled photons can be considered hyper-entangled—that is, entangled in two fully separable DoFs. The signal and idler photons from the first three comb line pairs are then selected and separated with a commercial pulse shaper325, as shown inFIG.5a. Now that the time bins and frequency bins are all generated in the state preparation stage, the idler photons are sent to the single photon detector340bto be used as heralding photons, and the signal photons are what carry the two-qudits in the three time bins {|0t, |1t, |2t} and frequency bins {|0f, |1f, |2f}. This procedure allows preparation of any time-bin/frequency-bin product state |mt|nf(m, n=0,1,2) of the full computational basis set. In principle, arbitrary time-frequency superposition states can also be heralded in this setup, by first sending the idler photon through a combination of time- or frequency-bin interferometers prior to detection in the temporal and spectral eigenbases. This more general case would permit the preparation of any two qudit state.

In an exemplary operation using the CINC gate330a, the X gate355is applied to the time-bin qudit only when the frequency qudit is in the state |2f. This two-qudit gate, along with arbitrary single-qudit gates, which, as noted above, can be formed from single-qudit X and Z operations, complete a universal set for any quantum operation. To implement this gate, the frequency bin |2fis separated from the other two frequency bins, for example, using a dense wavelength-division multiplexing (DWDM) filter352. The DWDM filter352routes the frequency bin |2fto the path354aand thus to the time-bin X gate355. The DWDM filter352routes the frequency bins {|0f, |1f} to the other path354b, where no operation happens on those other two frequency bins. The combiner DWDM360recombines the frequency bins with zero relative delay to complete the two-qudit gate operation.

To measure the transformation matrix of this gate in the computational basis, the first intensity modulator315and the pulse shaper325, respectively, were used to prepare the input state in each of the 9 combinations of single time bins and frequency bins. The signal counts were then recorded in all possible output time-bin/frequency-bin pairs, conditioned on detection of a particular idler time frequency mode, by inserting three different DWDMs in the path of the signal photons to pick different frequency bins (the DWDM right before the single photon detector340a).

The measured transformation matrix is shown inFIG.5b, with accidental-subtracted fidelityC, =0.90±0.01. The accidentals were subtracted in the transformation matrices, and the coincidence to accidentals ratio was ˜3.7 in the CINC. As shown inFIG.5b, the time/frequency pairs with the frequency bin |2f, have been changed within the gate330a.

In the experimental setup as shown inFIG.5a, the same techniques used for the CINC gate330awere also used to measure the transfer matrix shown of a SUM gate330b, with the results shown inFIG.5c, withC=0.92±0.01. In this embodiment, the logic gate330is the SUM gate330b, which may suitably comprise a chirped fiber Bragg grating (CFBG)372and a time-bin X gate374. Similar to the X gate ofFIG.4a, the X gate374includes a Mach-Zehnder modulator MZM374a, an interferometric structure374band PZT374c. The SUM gate330b, which can be thought of as a generalized controlled-NOT gate, adds the value of the control qudit to the value of the target qudit, modulo 3. Thus, in the SUM gate330b, the time bins associated with |0fare not delayed, the time bins associated with |1fexperience a cyclic shift by 1 slot, and the time bins corresponding to |2fgo through a cyclic shift of 2 slots. To delay the time bins dependent of their frequencies, the CFBG372induces a dispersion of −2 ns/nm on the photons, imparting 6-ns (1-bin) and 12-ns (2-bin) delays for the temporal modes associated with |1fand |2f, respectively, as required for the SUM operation. However, this delay is linear—not cyclic—so that some of the time bins are pushed outside of the computational space, to modes |3tand |4t. The MZM374aseparates the time bins that fall outside of the computational space (|3tand |4t) from the computational space time bins (|0t, |1tand |2t). Returning these bins to overlap with the necessary |0tand |1tslots can be achieved using principles identical to the time-bin X gate205with a relative delay of three bins, hence the inclusion of the X gate374.

The fact that this SUM gate330bis implemented with qudits in a single step potentially reduces the complexity and depth of quantum circuits in all the algorithms that require an addition operation. We note that to enhance computational capabilities, it would be valuable to also develop two qudit operations where instead time bins are the control qudit and frequency bins the target qudit which would then require active frequency shifting conditioned on time bins.

To show the ability of the techniques disclosed herein to operate on extremely large Hilbert spaces, the dimensions of the qudits are extended by encoding two 16-dimensional quantum states in the time and frequency DoFs of a single photon. For this demonstration, because more time bins and smaller frequency spacing between modes are desired, a broadband source of time-frequency entangled photons is used instead of the microring320with fixed frequency spacings. A 773 nm CW laser is shined on a periodically poled lithium niobate (PPLN) crystal, generating entangled photons with a bandwidth of ˜5 THz. as discussed in Imany, P., Odele, O. D., Jaramillo-Villegas, J. A., Leaird, D. E. & Weiner, A. M. “Characterization of coherent quantum frequency combs using electro-optic phase modulation.”Phys. Rev. A97, (2018). 16 time bins are then carved with a full width at half maximum of ˜200 ps and 1.2 ns spacing between them, to generate the time-bin qudits. Then, a pulse shaper is used to carve out the frequency of these entangled photons to generate sixteen 22 GHz wide frequency bins on both the signal and idler side of the spectrum, each 75 GHz spaced from each other. Now that there are 16-dimensional qudits in both time and frequency, the signal photon (heralded by the idler photon) into the same SUM-gate structure330b. After the CFBG372, the time bins will spread to ˜300 ps due to their large bandwidth. This spreading can be reduced by using a smaller linewidth for our frequency modes, for example with a Fabry-Perot etalon. To verify the operation, different input two-qudit states were sent, chosen from one of 256 basis states, and the output was measured after the gate330b.

While this yields a total of 256×256 (216) computational input/output combinations to test, there were no active frequency shifting elements in the setup, so it is reasonable to assume that the frequency qudit remains unchanged through the operation. This is also enforced by the high extinction ratio of the pulse shaper (˜40 dB), which blocks unwanted frequency bins. This allows one to focus on results in the sixteen 16×16 transfer matrices measured inFIGS.6A and6B(a subset with a total of 212input/output combinations). In each matrix, 16 different inputs with the same frequency and different time bins are sent into the SUM gate330band the output time bins are measured.

For this experiment, superconducting nanowire single photon detectors (SNSPDs) as the detectors340a,340bwere used, which allow the report of data without accidental subtraction. The average computational space fidelity for the whole process, with the assumption that frequencies do not leak into each other, can be calculated asC=0.9589±0.0005, which shows the high performance of our operation. This high fidelity is the result of the high extinction ratio of the intensity modulator used to carve the time bins (˜25 dB). To show the coherence of the SUM gate330b, the same setup performed a SUM operation on a three-dimensional input state,

❘"\[LeftBracketingBar]"ψ〉i⁢n=13⁢(❘"\[LeftBracketingBar]"0〉f+❘"\[LeftBracketingBar]"1〉f+❘"\[LeftBracketingBar]"2〉f)⁢❘"\[LeftBracketingBar]"0〉t
which results in a state maximally entangled between time and frequency DoFs:

❘"\[LeftBracketingBar]"ψ〉out=13⁢(❘"\[LeftBracketingBar]"00〉ft+❘"\[LeftBracketingBar]"11〉ft+❘"\[LeftBracketingBar]"22〉ft).
To quantify the dimensionality of this entangled state, an entanglement certification measure called entanglement of formation (Eof) was used, as discussed in Martin, A. et al. “Quantifying Photonic High-Dimensional Entanglement.”Phys. Rev. Lett.118, (2017), and Tiranov, A. et al. “Quantification of multidimensional entanglement stored in a crystal.”Phys. Rev. A96, (2017).

Eof≥1.19±0.12 ebits were experimentally obtained, where 1 ebit corresponds to a maximally entangled pair of qubits, while 1.585 ebits represents the maximum for two three-dimensional parties (see Methods); in exceeding the qubit limit, our state thus possesses true high-dimensional entanglement.

One of the most crucial challenges towards optical quantum operations is the lack of on-demand photon sources. Therefore, it is interesting to consider the techniques described herein for application to quantum communication and networking, for which operations with just a few qudits have potential impact. A gate similar to the SUM gate330bis the XOR gate, which subtracts the control qudit from the target and is a requirement for qudit teleportation protocols, as discussed in Roa, L., Delgado, A. & Fuentes-Guridi, I. “Optimal conclusive teleportation of quantum states.”Phys. Rev. A At. Mol. Opt. Phys.68, 6 (2003). Since teleportation of quantum states is possible using different degrees of freedom of an entangled photon pair, a single photon two-qudit gate such as those ofFIGS.4aand5acould be applied directly for teleporting high-dimensional states. Specifically, the XOR gate can be demonstrated by using positive dispersion and reconfiguring the switching in the SUM gate, or in the three-dimensional case, by simply relabeling the frequency bins |0f→|2fand |2f→|0fand performing the same process as the SUM operation.

Additionally, these two-qudit gates can be used for the purpose of beating the channel capacity limit for standard superdense coding for high-dimensional entangled states. The general concept of beating the channel capacity limit is discussed in Barreiro, J. T., Wei, T. C. & Kwiat, P. G. “Beating the channel capacity limit for linear photonic superdense coding.”Nat. Phys.4, 282-286 (2008). In such quantum communications applications for the two-qudit gates, a modest number of state manipulations brings potential impact.

Variants of the demonstrated SUM gate330bcan also be used to produce high-dimensional Greenberger-Horne-Zeilinger (GHZ) states. GHZ states consist of more than two parties, entangled with each other in a way that measurement of one party in the computational basis determines the state of all the other parties. Such properties of GHZ states are discussed, for example, in Pan, J. W., Bouwmeester, D., Daniell, M., Weinfurter, H. & Zellinger, A. “Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement.”Nature403, 515-519 (2000), which is incorporated herein by reference.

Such states have many interesting applications such as confirmation of Bell's theorem without inequalities, quantum secret sharing, as discussed in Hillery, M., Buz̆ek, V. & Berthiaume, A. “Quantum secret sharing.”Phys. Rev. A59, 1829 (1999), and open destination quantum teleportation, as discussed in Zhao, Z. et al. “Experimental demonstration of five-photon entanglement and open-destination teleportation.”Nature430, 54-58 (2004). It has been only recently that these states were demonstrated in more than two dimensions, where a three-dimensional three-party GHZ state was realized using the orbital angular momentum of optical states.

In the embodiment described herein, the photons are encoded in multiple time bins and frequency bins. To this end, the frequency spacing between different modes (Δf) and the time-bin spacing (Δt) are chosen such that they far exceed the Fourier transform limit (i.e., ΔfΔt>>1). As such, it is possible to manipulate the time and frequency DoFs independently in a hyper-encoding fashion, using concepts developed in time-division and wavelength-division multiplexing, respectively. Such techniques are discussed in Fang, W.-T. et al. “Towards high-capacity quantum communications by combining wavelength-and timedivision multiplexing technologies.”arXiv Prepr. arXiv1803.02003 (2018), and Humphreys, P. C. et al. “Continuous-variable quantum computing in optical time-frequency modes using quantum memories.”Phys. Rev. Lett.113, (2014), both of which are incorporated herein by reference. As a result, each time-frequency mode pair constitutes a well-defined entity, or plaquette, which is sufficiently separated from its neighbors to provide stable encoding.

FIG.7ashows a flow diagram of another method that generates multiparty states in the frequency and time degrees of freedom. In step402, first photon and second photon are generated, each having at least two frequency bins, (preferably three or more) and at least one time bin. The first and second photons are entangled with each other. In this embodiment, the first and second photon only have a single time bin in step402.

Thereafter, in step404, a frequency dependent time shift, for example, a time delay, is performed on each of the first and second photons to form at least a second time bin, and entangle (i.e. make nonseparable) the time and frequency bins within each photon to create four-party Greenberger-Horne-Zeilinger (GHZ) states. If the initial photons are entangled in more than two frequency bins, the final state will be a four-party multidimensional GHZ state. Step404may be carried out by the novel quantum gate discussed herebelow. The resultant GHZ states consist of more than two parties, entangled with each other in a way that measurement of one party in the computational basis determines the state of all the other parties. Such properties of GHZ states are discussed, for example, in Pan, J. W., Bouwmeester, D., Daniell, M., Weinfurter, H. & Zellinger, A. “Experimental test of quantum nonlocality in three-photon Greenberger-Horne-Zeilinger entanglement.”Nature403, 515-519 (2000), which is incorporated herein by reference.

In the present disclosure the SUM gate and the large dimensionality of time-frequency states facilitate the generation of a four-party GHZ state with 32 dimensions in each DoF. The starting state is expressed as the state,

❘"\[LeftBracketingBar]"ψ〉i⁢n=132⁢❘"\[LeftBracketingBar]"0,0〉ts⁢ti⁢∑m=03⁢1⁢❘"\[LeftBracketingBar]"m,m〉fs⁢fi
which means both the first photon and the second photon are initialized (step405) in the first time-bin state and are maximally entangled in the frequency domain. Then, deterministic SUM gates operate (step407) separately on both the first and second photons, resulting in a four-party GHZ state of the form

❘"\[LeftBracketingBar]"ψ〉out=132⁢Σm=03⁢1⁢❘"\[LeftBracketingBar]"m,m,m,m〉fs⁢ts⁢fi⁢ti.

Such four-party GHZ states are useful, for example, in quantum key distribution and other quantum communication protocols, as there are two highly entangled photons. Accordingly, in step406, the first photon is transmitted to a location that is different from the location of the second photon. Thereafter, in step408, a deterministic quantum operation is performed at least on the first photon, for example, consistent with quantum key distribution. As noted inFIG.3, the first photon, having multiple frequency and time bins that are inseparable or entangled with each other, is suited for deterministic operations.

FIG.7bshows a schematic block diagram of an apparatus for generating large-scale optical entangled states using high-dimensional quantum logic, and which may carry out at least steps402and404ofFIG.7a. In general, the apparatus410includes a generator412and a SUM gate424. The generator412, combined with the pulse shaper420is configured to generate a signal (first) photon and an idler (second) photon such that the signal photon and idler photon are entangled in the frequency domain in two or more different frequency bins, and preferably more than three different frequency bins, and such that the signal photon and idler photon have the same time value. As will be discussed further below, the pulse shaper420can also separate the single and idler photons into different routes. As discussed above, methods of (and accompanying component elements for) generating signal and idler photons with entangled frequency states and one or more entangled time states are known. In one embodiment, the generator412includes a coherent light source such as a pump laser416, operably coupled to a lithium niobate crystal417, which is further coupled to an intensity modulator418. However, other suitable arrangements may be used. In one example, the order of the lithium niobate crystal417and intensity modulator418is reversed.

The SUM gate424in this configuration is a dispersion module424, for example, a chirped fiber Bragg grating (CFBG). As discussed above, the SUM gate424is an optical gate configured to perform a deterministic action on a respective one of the signal and idler photons to entangle (i.e., make inseparable) the time and frequency degrees of freedom in each photon. Before separating the photons with a pulse shaper420, a single CFBG can operate the SUM gate424on both signal and idler photons even before generating the frequency bins with the pulse shaper420.

In operation, the generator412, combined with the pulse shaper420, operates to generate signal and idler photons that are highly entangled with each other in a plurality of frequency bins, and in a single time bin. The SUM gate424performs a deterministic action on each of the signal and idler photons to entangle (make inseparable) the time and frequency degrees of freedom in each photon, thereby creating four-party Greenberger-Horne-Zeilinger states which can be multidimensional, having wave functions characterized by

❘"\[LeftBracketingBar]"ψ〉out=1√d⁢Σm=0d-1⁢❘"\[LeftBracketingBar]"m,m,m,m〉fs⁢ts⁢fi⁢tiwherein fsis the signal frequency party, fiis the idler frequency party, tsis the signal time party, and tiis the idler time party.

Since the initial state only consists of the first time bins, the dispersion module424does not shift any of the bins outside of the computational space; hence the interferometric structure used in the full SUM gate330bis not required when operating within this subspace. The GHZ state is measured in the computational basis. The pulse shaper420may then be used to separate the signal and idler photons, allowing for their different subsequent uses. Such subsequent uses may include transmitting the signal and idler photons to different locations and performing deterministic operations on either or both, as discussed above in connection withFIG.7a.

FIG.8ashows a plot of coincidences for all basis states in the set {|m, n, k, lfstsfiti; 0≤m, n, k, l≤31}. Only states whose four qudits match (i.e. |m, m, m, mfstsfiti) have high counts, as expected for a GHZ state.

In particular,FIG.8ashows the measurement of the four-party 32-dimensional GHZ state in the computational basis, generated in accordance with the method ofFIG.2and in the system ofFIG.7b. The states |m, nshown on the signal and idler axes correspond to frequency-bin m and time bin n. The large coincidence peaks exist only for states with the same time-bin and frequency-bin indices for both signal and idler (32peaks).FIGS.8band8cshow zoomed in 32×32 submatrices of the matrix shown in FIG.8a. Each submatrix shows coincidences for different signal and idler time bin indices for fixed signal and idler frequency bin indices.FIG.8bshows matched signal and idler frequency bins, where a large peak is observed for |16, 16, 16, 16fstsfiti.FIG.8cshows unmatched signal and idler frequency bins. The small peak evident at |16, 16, 23, 23fstsfitireflects additional accidentals (multiphoton pair events) at the time bins to which frequency bins |16f, and |23fiare shifted. The data are shown with accidentals subtracted (coincidence to accidentals ratio of ˜4).

It will be appreciated that full characterization of the state requires measurements in superposition bases as well, but due to the additional insertion loss associated with superposition measurements in time and frequency using interferometers and phase modulators, respectively, such projections were not measured in this experiment. It is anticipated that there would be dramatically lower insertion losses with on-chip components, which may make these measurements possible. Remarkably, the demonstrated GHZ state resides in a Hilbert space equivalent to that of 20 qubits, an impressive 1,048,576 (324) dimensions. The realization of such high-dimensional GHZ states indicates the potential of the time-frequency platform for near-term quantum technologies such as cluster-state quantum computation.

Hyper-entangled time-frequency entangled states, as opposed to other high-dimensional optical degrees of freedom like orbital angular momentum, can be generated in integrated on-chip sources, which have gained tremendous attention in recent years due to their low cost, room temperature operation, compatibility with CMOS foundries and the ability to be integrated with other optical components. Pulse shapers, phase modulators and MZMs can all be demonstrated on a chip, and a series of DWDMs and delay lines can be used to demonstrate an on-chip CFBG. In addition, demonstration of balanced and unbalanced interferometers on-chip reduces fast fluctuations, making stabilization easier, which is of considerable profit for the scalability of the scheme. These contributions can potentially lead to combining these sources with on-chip components designed for manipulation of these states, to create the whole process on an integrated circuit.

As discussed, above, high-dimensional optical states can open the door to deterministically carry out various quantum operations in relatively large Hilbert spaces, as well as having higher encoding efficiency in quantum communication protocols such as quantum key distribution and quantum teleportation. Experiments disclosed herein have demonstrated deterministic single- and two-qudit gates using the time and frequency degrees of freedom of a single photon for encoding—operating on up to 256 (28)-dimensional Hilbert spaces—and carried out these gates with a high computational space fidelity. The application of such two-qudit gates in near-term quantum computation is possible by using them to realize a GHZ state of four parties with 32 dimensions each, corresponding to a Hilbert space of more than one million modes. Such deterministic quantum gates add significant value to the photonic platform for quantum information processing and have direct application in, e.g., simulation of quantum many-body physics.

Methods

For the time-bin single qudit X gate205shown inFIG.4a, we split the experimental setup200in three stages: state preparation215, X gate205operation and state measurement240. For the state preparation215, we use an Agilent 81645A CW laser210tuned to 1553.9 nm and send it into an intensity modulator220(˜4 dB insertion loss) and phase modulator225(˜3 dB insertion loss), both manufactured by EOSpace, which are used to create the time bins and control their relative phases, respectively. To implement the X gate205, we used an MZM230with two complementary outputs (˜4 dB insertion loss), also manufactured by EOSpace. We also use a piezo-based fiber phase shifter235(General Photonics FPS-001) to control the phase difference between the two paths following the MZM230. Then a 2×2 fiber coupler250is used to merge the branches. For the state measurement240, we used 1-bin and 2-bin delay interferometers260,265implemented with 2×2 3 dB fiber couplers and additional piezo-based fiber phase shifters260a,260b.

For the time-bin X gate and computational-basis measurements of three-dimensional two-qudit gates, gated InGaAs single photon detectors (Aurea Technologies SPD_AT_M2)245were used. For the rest of the measurements, we used superconducting nanowire single photon detectors (Quantum Opus). To measure the arrival times of the photons on the single photon detectors, a time interval analyzer (PicoQuant HydraHarp 400) is used. The stabilization of the interferometers is done by sending a CW laser at 1550.9 nm in the backwards direction and feeding the output power into a computer based feedback loop to maintain the phase. To stabilize the X gate205, we use a similar scheme with an additional circulator at the input of the gate (not shown in the figures) to retrieve the optical power in the backwards direction. The signal applied to the intensity modulators and phase modulator, as well as the trigger and synchronization signal of the single photon detector and time interval analyzer, are generated by an electronic arbitrary waveform generator Tektronix AWG7122B and adjusted to the proper level by linear amplifiers.

To assess the performance of our one-and two-qudit quantum gates, we first focus on the computational-basis fidelityC—one example of a so-called “classical” fidelity in the literature, such as in De Greve, K. et al. “Complete tomography of a high-fidelity solid-state entangled spin-photon qubit pair.”Nat. Commun.4, 2228 (2013) 49. Defining |n(n=0, 1, . . . , N−1) as the set of all computational basis states and |uVas the corresponding output states for a perfect operation, we have the fidelity

ℱc=1N⁢∑n=0N-1p⁡(un❘n)(1)where p(uN|n) is the probability of measuring the output state |ungiven an input of |n. In the operations considered here, the ideal output states |unare members of the computational basis as well, so there is no need to measure temporal or spectral superpositions in determination ofc. Given the measured counts, we retrieve the N conditional probability distributions via Bayesian mean estimation (BME) where our model assumes that each set of count outcomes (after accidentals subtraction) follows a multinomial distribution with to-be-determined probabilities; for simplicity, we take the prior distributions as uniform (equal weights for all outcomes). We then compute the mean and standard deviation of each value p(un|n) and sum them to arrive atc. Specifically, if Cun|nsignifies the counts measured for outcome un, and Ctot|nthe total counts over all outcomes (both for a given input state |n), BME predicts:

p⁡(un❘n)=1+Cun❘nN+Ctot❘n±1+Cun❘n(N+Ctot❘n)2⁢N+Ctot❘n-Cun❘n-1N+Ctot❘n+1(2)where the standard deviation in the estimate is used for the error. Since the probabilities here each actually come from N different distributions, we estimate the total error inCby adding these constituent errors in quadrature. Explicitly, we findC=0.996±0.001 for the X gate, 0.90±0.01 for the CINC operation, 0.92±0.01 for the 3×3 SUM gate, andC=0.9589±0.0005 for the 16×16 SUM gate. The reduction inCfor the two-qudit gates is due in large part to the fewer total counts in these cases, from our use of heralded single photons rather than a weak coherent state. As seen by the presence of N in the denominator of Eq. (2), even when Cun|n=Ctot|n, the estimate (un|n) is not unity unless Ctot|n>>N. In our experiments, the two-qudit tests have only ˜100-300 total counts per input computational basis state for the 9×9 matrices (with N=9) and ˜500-800 counts per input state for the 16×16 matrices (with N=16), thereby effectively bounding the maximum p(un|n) and, by extension, fidelityC. This behavior is actually a strength of BME, though, as it ensures that we have a conservative estimate of the fidelity that is justified by the total amount of data acquired. as discussed in Blume-Kohout, R. “Optimal, reliable estimation of quantum states.”New J. Phys.12, 043034 (2010).

While extremely useful for initial characterization, however, the computational-basis fidelity above provides no information on phase coherence. On the other hand, process tomography would offer a complete quantification of the quantum gate. Yet due to the challenging experimental complexity involved in quantum process tomography, here we choose a much simpler test which—while limited—nonetheless offers strong evidence for the coherence of our time-bin X gate. To begin with, note that all three-dimensional quantum processes can be expressed in terms of the nine Weyl operations, as discussed in Bertlmann, R. A. & Krammer, P. “Bloch vectors for qudits.”J. Phys. A Math. Theor.41, 235303 (2008):

U0=I=(100010001),(3)U1=X=(001100010),U2=X2=(010001100)U3=Z=(1000ei⁢2⁢π3000e-i⁢2⁢π3),U4=ZX=(001ei⁢2⁢π3000e-i⁢2⁢π30),U5=ZX2=(01000ei⁢2⁢π3e-i⁢2⁢π300),U6=Z2=(1000e-i⁢2⁢π3000ei⁢2⁢π3),U7=Z2⁢X=(001e-i⁢2⁢π3000ei⁢2⁢π30),U8=Z2⁢X2=(01000e-i⁢2⁢π3ei⁢2⁢π300),

The quantum process itself is a completely positive map ε (see O'Brien, J. L. et al. Quantum Process Tomography of a Controlled-NOT Gate.Phys. Rev. Lett.93, 80502 (2004), which for a given input density matrix ρinoutputs the state

ρout=ε⁡(ρi⁢n)=∑m,n=08χmn⁢Um⁢ρi⁢n⁢Un†(4)

The process matrix with elements χmnuniquely describes the operation. The ideal three-bin X gate with process matrix χχ. has only one nonzero value, [χχ]11=1. To compare to this ideal, we assume the actual operation consists of a perfect X gate plus depolarizing (white) noise. In this case we have a total operation modeled as

ρout=λ⁢U1⁢ρi⁢n⁢U1†+(1-λ)33(5)whose process matrix we take to be

xN=λXx+1-λ99,
which can be calculated by using

3=13⁢Σn=08⁢Un⁢ρi⁢n⁢Un†19.
if we then assume a pure input superposition state ρin=|ψinψin|, where |ψin∝|0t+eiØ|1t+e2iØ|2t, and measure the projection onto the output |ψout∝|0t+|1t+|2t(as inFIG.2c), λ can be estimated from the interference visibility V as:

λ≃2⁢V3-V(6)and the process fidelity is then given by:

ℱP=Tr⁡(XX⁢XN)=[XN]11=1+8⁢λ9=1+5⁢V9-3⁢V=0.92±0.01(7)as discussed in the main text.

To show the coherence of our SUM gate330bofFIG.5a, we generate an input state in the signal photon which is in time-bin |0tand an equi-amplitude superposition in frequency

❘"\[LeftBracketingBar]"ψ〉i⁢n=13⁢(❘"\[LeftBracketingBar]"0〉f+❘"\[LeftBracketingBar]"1〉f+❘"\[LeftBracketingBar]"2〉f)⁢❘"\[LeftBracketingBar]"0〉t.
After passing this state through the SUM gate330b, the time-bin state of the photon is shifted based on the frequency, leaving us with a maximally entangled state

❘"\[LeftBracketingBar]"ψ〉out=13⁢(❘"\[LeftBracketingBar]"00〉f⁢t+❘"\[LeftBracketingBar]"11〉f⁢t+❘"\[LeftBracketingBar]"22〉f⁢t).
We note that since we are starting with time-bin zero, the time bins will not fall out of the computational space; therefore, the interferometric structure (e.g. elements260,265ofFIG.4a) is not needed for the SUM gate330band the dispersion module alone can do the operation. This saves us the extra insertion loss of the interferometers, which is an important parameter due to the low photon pair rate on the detectors in this particular experiment. To measure the 3-dimensional entanglement in |ψout, we must vary the phases of different signal frequency bins and time bins with a pulse shaper and phase modulator, respectively. To observe the effect of this phase sweep with our relatively slow single-photon detectors (with ˜100 ps jitter), an indistinguishable projection of all three time bins and frequency bins should be created.

FIG.9ashows schematic block diagram of an experimental set-up of a SUM gate520that does not include an interferometer. Specifically,FIG.9ashows a set up500, having spontaneous parametric down conversion source505, a pulse shaper510, an intensity modulator525, a dispersion module520, a phase modulator515, and a state projection stage530. The state projection stage530includes another dispersion module535, two additional phase modulators540,545, and two further pulse shapers550,555. The pulse shaper510diverts the signal photon and idler photon to paths510a,510brespectively. The path510aincludes the intensity modulator515, the dispersion module520(the sum gate), and phase modulator525, and the dispersion module535, the phase modulator540and pulse shaper550. The dispersion module520is a −2 ns/nm dispersion module, and the dispersion module535is a +2 ns/nm dispersion module. The heralding path510bincludes the phase modulator545and the pulse shaper555.

In general, the time bins can be projected on an indistinguishable state by using a cascade of interferometers, such as the interferometers240ofFIG.4a. However, in the experimental set-up500, it is simpler to use a dispersion module535with opposite dispersion to that of the module520used in the SUM gate to perform the same projection. After the dispersion module535, the frequency bins are then projected on an indistinguishable state using the phase modulator540and pulse shaper550to mix the frequencies—a technique used previously in Imany, P. et al. 50-GHz-spaced comb of high-dimensional frequency-bin entangled photons from an on-chip silicon nitride microresonator.Opt. Express26, 1825-1840 (2018), which is incorporated herein by reference.

We note that our measurements on the signal photons are conditioned on heralding by idler frequency superposition states. To measure the interference between different signal frequency bins, the idler photons too have to be projected on an indistinguishable frequency bin using the phase modulator545and pulse shaper555. This projection guarantees that detection of an idler photon does not give us any information on the frequency of the signal photon. Unlike prior experiments, however, the phases of the idler frequency bins are held constant; only the phases of the signal frequency and time bins are varied. This is in contrast to experiments in Imany, P. et al. comb of high-dimensional frequency-bin entangled photons from an on-chip silicon nitride microresonator.Opt. Express26, 1825-1840 (2018), where the phases of both signal and idler frequency bins were varied.

In the experiment, three-dimensional interference measurements were not possible since mixing all three frequencies together adds extra projection loss, which we cannot afford. Therefore, we vary the phases of different time bins and frequency bins to measure two-dimensional interference patterns between all three time bins and frequency bins (FIG.9c). Using the visibilities of these interference patterns along with a joint spectral intensity (JSI) measurement (FIG.9b) can give us a lower bound on the amount of entanglement present in our system by measuring entanglement of formation. (See Tiranov, A. et al. “Quantification of multidimensional entanglement stored in a crystal.”Phys. Rev. A96, (2017), and Barreiro, J. T., Wei, T. C. & Kwiat, P. G. “Beating the channel capacity limit for linear photonic superdense coding.”Nat. Phys.4, 282-286 (2008). The JSI denotes the correlations between the time bins and frequency bins of a signal photon heralded by an idler photon in its computational basis.

The same time-bin and frequency-bin spacings (1.2 ns, 75 GHz) as the 16-dimensional SUM gate experiment are used for these measurements. We note that in this experiment, the IM515was placed only on the signal photons' route to avoid its insertion loss on the idler photons.FIG.9bshows a graph of the Joint spectral intensity of the three-dimensional entangled state. The accidentals were subtracted in this measurement, with a coincidence to accidentals ratio of about 30.

FIG.9cshows graphs of two dimensional interference patterns showing the coherence between all three time-frequency modes of the entangled state. The frequency-bin and time-bin phases are varied using PS1and PM1, respectively. Both phases are swept together from 0 to π, for a total phase sweep from 0 to 2π. The data are shown with accidentals subtracted and coincidence to accidentals ratio of about 1. Since projection of frequency bins 0 and 2 on an indistinguishable frequency bin undergoes more projection loss, the coincidences between modes 0 and 2 were measured in 10 minutes.

The measurement was done using the same experimental setup used inFIG.9awithout the equipment used for sweeping the phase of different signal time bins and projection measurements. For this measurement, the idler photons were detected after PS1, and the signal photons were detected right after the SUM gate. Having the JSI measurement and the two-dimensional interference visibilities in hand, we have all the data needed to calculate the entanglement of formation in our system, which can be expressed as:where

Eof≥-log2(1-B22)(8)where⁢B=2❘"\[LeftBracketingBar]"C❘"\[RightBracketingBar]"⁢(∑(j,k)∈Cj<k❘"\[LeftBracketingBar]"〈j,j❘"\[RightBracketingBar]"⁢ρ⁢❘"\[LeftBracketingBar]"k,k〉❘"\[RightBracketingBar]"⁢-〈j,k⁢❘"\[LeftBracketingBar]"ρ⁢❘"\[LeftBracketingBar]"j,k〉⁢〈k,j❘"\[RightBracketingBar]"⁢ρ⁢❘"\[LeftBracketingBar]"k,j〉)(9)

Here, C is the number of indices (j, k) used in the sum. This measurement is useful when we do not have access to all the elements of the density matrix.j, j|ρ|k, k (j≠k)elements indicate the coherence between modes j and k, and can be lower-bounded using the two-dimensional visibilities. The termsj, k|ρ|j, kcan be calculated using the elements of the JSI. Using these values, we measure Eof≥1.19±0.12 ebits, which indicates greater than two dimensional entanglement in our bipartite system, more than one standard deviation away from the threshold. To generate the 32-dimensional four-party GHZ state, the signal and idler go through the same dispersion module (−2 ns/nm). After dispersion, the signal frequency bins farther away from the center of the spectrum are delayed more, but the idler frequency bins are delayed less as we move farther away from the center. In order to write the GHZ state in the form

❘"\[LeftBracketingBar]"ψ〉out=132⁢Σm=03⁢1⁢❘"\[LeftBracketingBar]"m,m,m,m〉fs⁢ts⁢fi⁢ti,
we label the signal time bins after dispersion 0 to 31 starting from earlier time bins (time bin 0 the earliest, time bin 31 the latest), while on the idlers, we label the time bins such that the earliest time bin is 31 and the latest time bin is 0. Another choice would be to send signal and idler through separate modules with equal but opposite dispersion, in which case we would use identical time labeling. To measure the state illustrated inFIG.5, we individually measured coincidences for the 32 different settings of both signal and idler frequency bins (32×32 measurements). For each of these measurements, we used our event timer to assign signal and idler time bins for each coincidence, which results in a 32×32 submatrix for each signal-idler frequency setting. Therefore, we have 324measurements in total. Two of the 32×32 time-bin submatrices are shown inFIG.8b,c.