Abstract
Controlling and programming quantum devices to process quantum information by the unit of quantum dit, i.e., qudit, provides the possibilities for noiseresilient quantum communications, delicate quantum molecular simulations, and efficient quantum computations, showing great potential to enhance the capabilities of qubitbased quantum technologies. Here, we report a programmable quditbased quantum processor in siliconphotonic integrated circuits and demonstrate its enhancement of quantum computational parallelism. The processor monolithically integrates all the key functionalities and capabilities of initialisation, manipulation, and measurement of the two quantum quart (ququart) states and multivalue quantumcontrolled logic gates with highlevel fidelities. By reprogramming the configuration of the processor, we implemented the most basic quantum Fourier transform algorithms, all in quaternary, to benchmark the enhancement of quantum parallelism using qudits, which include generalised DeutschJozsa and BernsteinVazirani algorithms, quaternary phase estimation and fast factorization algorithms. The monolithic integration and high programmability have allowed the implementations of more than one million highfidelity preparations, operations and projections of qudit states in the processor. Our work shows an integrated photonic quantum technology for quditbased quantum computing with enhanced capacity, accuracy, and efficiency, which could lead to the acceleration of building a largescale quantum computer.
Similar content being viewed by others
Introduction
Natural quantum matters store rich multidimensional quantum information in a superposition of more than two electronic or mechanical modes. Engineering artificial multilevel quantum devices to mimic nature may allow fundamental innovations and technological advances. Recently, though the stateoftheart qubitbased quantum technologies have demonstrated revolutionary milestones, e.g., loopholefree Bell tests^{1,2}, satelliterelayed quantum communications^{3,4} and quantum computational advantages^{5,6}, quditbased quantum technologies might be able to further enhance quantum capabilities as they are intrinsically consistent with our natural quantum systems. For example, entangled qudit states can strengthen the Bell nonlocality^{7} and moderate the detection loophole^{8}; distributing qudit states allows highcapacity noiseresilient quantum cryptography^{9,10,11}; by mapping Hamiltonians into multilevel quantum devices, it can provide a direct solution for quantum simulations of complex molecular and physical systems^{12,13,14,15,16}; more importantly, universal quantum computation with qudits is possible in both of the circuit models^{17} and measurementbased models^{18,19}, requires less resource overhead in quantum error correction^{20,21}, and can improve the execution of quantum algorithms^{22,23}. Heuristically, the exponential speedup of many quantum algorithms is enabled by the quantum parallel evaluation of a function f(x) for all input x values simultaneously, as \({\sum }_{x}\leftx\right\rangle \leftf(x)\right\rangle \), where the x input string is represented by a superposition of quantum states. The adoption of qudit as the basic quantum information unit in processing quantum algorithms^{24} offers enhanced computational capacity that is represented by the size of the Hilbert space of d^{n}, where n is the number of qudits and d is the local size of each qudit. Moreover, it can lead to higher computational accuracy for example in implementing quantum Fourier transform algorithms such as Shor’s fast factorisation^{25} and phase estimation^{26}, in which the computational accuracy is determined by the size of auxiliary qudits. Processing the Kitaev’s version of quantum Fourier algorithms^{26,27,28,29} with qudits may allow further speedup of quantum computing. These unique capabilities have strongly prompted the development of quditbased quantum computing in universal models^{17,18,19,30,31}, and very recently in experimental controls of qudit states and logic gates in photonics^{32}, solidstate^{15}, trapped ion^{33}, and superconducting^{34} platforms. In particular, photons are intrinsically multidimensional^{35}, enabling flexible and reliable encoding of qudits with their different degreesoffreedom, e.g, path^{36,37}, frequency^{38,39}, spatial mode^{40,41} and temporal mode^{11,42}. Advances in the control of quantum photonic devices have recently allowed remarkable experimental progress. For example, multidimensional GreenbergerHorneZeilinger (GHZ) states and cluster states prepared in the frequencybins and timebins of two photons generated in a single microring resonator^{43,44}, have firstly shown enhancement in quantum computation by providing increased quantum resources and higher noise robustness compared to the qubit counterparts; An integrated photonic chip for the generation, manipulation and measurement of twophoton multidimensional Bell states has been demonstrated^{36}, while the scaling capability has been verified by the generation of multiphoton multidimensional GHZ states^{45,46}, and the realisation of singlequdit quantum teleportation^{47,48}. Despite of these remarkable development of multidimensional quantum photonic technologies that mainly focus on the preparation and control of qudit states and gates, a monolithically integrated quantum device that is able to initialise, manipulate and analyze qudit states and gates is lacking. Furthermore, the programmability of quantum hardware presents the major enabling capability of quantum computing technologies. For example, several milestones in qubitbased quantum computing have been all realised in programmable quantum devices of photons^{49,50}, trapped ions^{51,52}, superconductors^{5,6} and semiconductors^{53}. However, limited to the best to our knowledge, such a quditbased quantum computing device that can be fully reconfigured and reprogrammed to implement different tasks has not been realised to date, in any quantum system. Likely, it requires an integrated platform^{35,54,55}, capable of initialising, manipulating and measuring qudit states and gates, in a fully controllable and highly programmable manner. Realising a programmable quditbased quantum processor therefore presents a significant step to transition the technological advances of controlling qudit states and logic gates to the implementations of quantum tasks and quantum computational algorithms, in dary.
In this work, we demonstrate a programmable quditbased quantum processing unit (dQPU) on a largescale siliconphotonic quantum chip. The initialisation, manipulation and measurement of arbitrary singlequdit and twoqudit states, and multivalue quantumcontrolled logic gates can be implemented on the single dQPU chip. Such a fully monolithic integration of all necessary functionalities allows the implementation of a topdown hierarchy of programmable quditbased quantum computation, as shown in Fig. 1. Different quantum tasks and computational algorithms are implemented, all in quaternary, by recompiling the qudit logic circuit in the software level, and then executing the circuit by reprograming the configurations of the dQPU chip in the hardware level. We then benchmark the enhancement of quantum computational parallelism, by performing the generalised DeutschJozsa and BernsteinVazirani algorithms, quaternary phase estimation and order finding algorithms. Our results show a proofofprinciple demonstration of quditbased quantum computer with integrated optics, that allows improvement of the capacity, accuracy and efficiency of quantum computing.
Results
Scheme of multiqudit quantum processor
Figure 2 shows the core of a multiqudit processor, i.e, the multiqudit multivalue controlled logic gate, which is realised by the following three steps: generation of the multiphoton multidimensional GreenbergerHorneZeilinger entangled state \({\left{{\mbox{GHZ}}}\right\rangle }_{n+1,d}\)^{45,46}, which enables the entangling operations between the multiqudit states; Hilbert space expansion of each qudit in yregister to form an entire space of d^{2n}, that locally allows individual and arbitrary singlequdit operations^{56}; coherent compression of the entire state back to a d^{n} space^{57}. These sequences of operations result in a multiqudit multivalue controlledunitary (MVCU) gate as \(\frac{1}{\sqrt{d}}\mathop{\sum }\nolimits_{j = 0}^{d1}\left{k}_{j}\right\rangle \otimes \mathop{\prod }\nolimits_{i = 1}^{n}{O}_{i,j}{\left\phi \right\rangle }_{i}\), where \(\left{k}_{j}\right\rangle \) in the auxiliary xregister presents the logical state in the jth mode (for simplicity it is denoted as \(\leftj\right\rangle \)), and O_{i,j} in the data yregister refers to an arbitrary local operation on the qudit state \({\left\phi \right\rangle }_{i}\) that is initialised by the _{Pi} qudit generator. Such multiqudit MVCU gate works with a (1/d) success probability regardless of n (see Supplementary Note 3 and Supplementary Fig. 1). The quantum circuits in Fig. 2a, b provide a scheme of implementing multiqudit quantum Fourier algorithms in the scalable Kitaev’s framework^{26,27,28,29}.
Figure 2c illustrates the integrated photonic quantum circuits for a twoququart version of quditbased quantum processing unit (dQPU). It was fabricated in silicon using the complementary metaloxidesemiconductor (CMOS) process with the 248nm deep ultraviolet lithography (see a device image in Fig. 2d). The processor allows the generation of a pathencoded twoququart entangled state of \({\left{{\mbox{GHZ}}}\right\rangle }_{2,4}\) (i.e., the 4dimensional generalised Bell state of \({\left{{\mbox{Bell}}}\right\rangle }_{4}\)), by a coherent excitation of four integrated spontaneous fourwavemixing (SFWM) sources. It is followed by the sequences of processes of “space expansion–local operation–coherent compression" for the realisation of dQPU, see Fig. 2b. The dQPU chip monolithically integrates the core capabilities and functionalities, including arbitrary singleququart preparation (P), arbitrary twoququart MVCU operation (that presents a dary generalisation of twoqubit controlledunitary operation), and arbitrary singleququart measurement (M). Though onchip generation, manipulation and measurement of entangled qudit states have been reported^{36}, this work demonstrate the key abilities to initialize, manipulate, and analyze qudit states and gates in a fully reconfigurable and reprogrammable manner, providing a major technological advance for qudit quantum computing. In Fig. 2d it shows one of the largestscale programmable quantum photonic chip having 451 photonic components, including 116 reconfigurable phaseshifters (see their characterisations in Fig. 2c insets). The twophotons detection rate at the magnitude of 10^{3}/s was measured in the twoququart device, which is six orders higher than that in a fourqubit device (note the detection rate depends on the performance and loss of the quantum devices as well as their pumping and measurement apparatuses)^{58}. Details of device fabrication, state evolution and experimental setup are provided in Supplementary Notes 1 and 3.
Characterisation of dary multivalue controlledunitary gates
Before reporting experimental results, we first define classical statistic fidelity (F_{c}) and quantum state (process) fidelity (F_{q}), used in this work to quantify the qudit states, logic gates and algorithm implementations. The F_{c} is defined as \({({\sum }_{i}\sqrt{{p}_{i}{q}_{i}})}^{2}\), where _{pi}, q_{i} are theoretical and measured distributions, respectively; the state F_{q} is defined as \({({{\mbox{Tr}}}[\sqrt{\sqrt{{\rho }_{0}}\cdot \rho \cdot \sqrt{{\rho }_{0}}}])}^{2}\), where ρ_{0}, ρ are ideal and measured states, respectively; the process F_{q} is defined as Tr[χ_{0}χ], where χ_{0}, χ are ideal and reconstructed process matrices, respectively.
We first characterised the singleququart and twoququart logic gates. As examples, two singleququart gates are characterised: the generalised dlevel PauliX_{d} gate that is defined as \({X}_{d}\left{k}_{i}\right\rangle =\left{k}_{(i{\oplus }_{d}1)}\right\rangle \) where ⊕ _{d} is addition module of d, and the dlevel quantum Fourier gate F_{d} that transforms the computational basis of \(\left{k}_{i}\right\rangle \) to the Fourier basis \(\left{f}_{i}\right\rangle \) of \(\frac{1}{\sqrt{d}}\mathop{\sum }\nolimits_{j = 0}^{d1}{\omega }^{ij}\leftj\right\rangle \) where i, j ∈ N_{d} and ω = \(\,{{\mbox{exp}}}\,({\mathbb{i}}\frac{2\pi }{d})\). When d is two, they return to the standard Pauli and Fourier (Hadamard) gates for qubits. In Fig. 2c inset, it shows the measured mean F_{c} of 0.988(13) for the five X_{4} gates and 0.967(19) for the five F_{4} gates, where the values in parentheses are uncertainty from photon statistics. Next, we characterised the twoqudit entangling gate:
where O can be arbitrarily operated^{59} on the \(\leftx\right\rangle \) and \(\lefty\right\rangle \) registers. Notably, the MVCU gate presents a coherent entanglement between the auxiliary xregister and the data yregister. The processing of dary quantum algorithms relies on the multiple path interference in the ddimensional Fourier gate to obtain the desired solutions. Such coherent superposition of qudits ensures quantum parallelism, that is function evaluations for multiple inputs are executed in parallel. The MVCU is thus a core logic enabling the quantum parallel evaluation of the function. For example, as the dary generalisation of the CNOT gate^{24}, the MVCX_{d} gate allows the creation of a complete set of fourlevel Bell states \({\left{{\Psi }}\right\rangle }_{i,j}\) defined as \(\frac{1}{2}\mathop{\sum }\nolimits_{m = 0}^{3}{\omega }^{mi}\leftm\right\rangle \leftm{\oplus }_{d}j\right\rangle \), by inputing the \(\left{f}_{i}\right\rangle \otimes \left{k}_{j}\right\rangle \) states into the logic, i, j = 0,1,2,3. Figure 3a shows the reconstructed \({\left{{\Psi }}\right\rangle }_{12}\) state, and Fig. 3c shows measured F_{q} for the 16 Bell states with an averaged fidelity of 0.967(31). The state matrices (ρ) represented as a linear combination of GellMann matrices were reconstructed by implementing compressed sensing quantum state tomography techniques^{60}. In addition, a fully product state was created in Fig. 3b, given an input of \(\left{f}_{0}{f}_{0}\right\rangle \). Figure 3d shows the experimental process matrix (χ) of the MVCX_{d} gate, by performing quantum process tomography with a full set of 256 state tomographic measurements^{61}, and a process fidelity F_{q} of 0.952 was obtained. We then characterised the MVCZ_{d} gate (Z_{d} is the generalised dlevel PauliZ_{d} gate) transforming \(\leftx\right\rangle \lefty\right\rangle \) to \(\leftx\right\rangle {\omega }^{xy}\lefty\right\rangle \), and the MVCH_{d} gate where H_{d} is the dlevel Hadamard gate with elements of \(\frac{1}{\sqrt{d}}{(1)}^{i\odot j}\) (i ⊙ j is the bitwise dot product, see Supplementary Note 2). Instead of performing full process tomography, we adopted an efficient characterisation by using complementary classical fidelity^{62}. Figure 3e–j show measured inputoutput truth tables and their classical fidelity (F_{c1}, F_{c2}) for the MVCU in two complementary {base I, base II}, from which the complementary classical fidelity is upper and lower bounded by [F_{c1} + F_{c2} − 1, Min(F_{c1}, F_{c2})].
Experimental implementation of dary Deutsch’s algorithms
The class of Deutsch’s algorithms well identify quantum parallelism. A generalised dary DeutschJozsa algorithm can determine whether a multivalue function f: {0, 1,..., d−1}^{n} → {0, 1,..., d − 1} is constant or balanced by a single query of a quantum oracle^{63}. Classically, it however requires d^{n−1} + 1 queries. The quantum circuit performing f(x) ⊕ _{d}y is shown in Fig. 4a. In the case of d = 2, it returns to the original binary DeutschJozsa^{64}. We implemented the ququart DeutschJozsa algorithm on the dQPU for the case of n = 1 and d = 4. Figure 4b–h show the measured probability distributions of the xregister in the computational basis, when the multivalue function is chosen as constant (see Fig. 4b) and balanced (see Fig. 4c–h), respectively. The dQPU thus determines whether f is constant or balanced, and the fidelity F_{c} of 0.967(2) was measured to quantify its success probability. Notably, the measured distributions in Fig. 4b, c, h, i are fully distinguishable. These imply an interesting capability of computing a close expression for an affine function f: A_{0} ⊕ A_{1}x_{1} ⊕ . . . ⊕ A_{n}x_{n}. That presents the dary generalisation of the BernsteinVazirani algorithm^{65}, whose task is to compute the dary coefficients A_{i}. The output state of the x register can be derived as \({\omega }^{{A}_{0}}\left{A}_{1},{A}_{2},...,{A}_{n}\right\rangle \), where the \(\left{A}_{1},{A}_{2},...,{A}_{n}\right\rangle \) state can be directly read out in its computational basis (A_{0} is lost as a global phase).
From the experimental results in Fig. 4b, c, h, i one can therefore determine the multivalue function with A_{1}= {0, 1, 2, 3}, respectively, by a single query of the oracle. Details of the generalisation of the Deutsch’s algorithms are provided in Supplementary Note 4.
Benchmarking of dary phase estimation and order finding
Quantum phase estimation and order finding are two of the most featured quantum Fourier transform ralgorithms, that are essential to molecular simulation^{66} and fast factorisation^{25}. Kitaev’s scalable implementation of both algorithms (in binary)^{26,27,28,29} has been reported in several leading quantum platforms^{67,68,69,70,71,72}. The remarkable idea is to replace the 2n qubits by a single qubit in the auxiliary xregister, but at the expense of repeating msequences of singlequbit measurement and singlequbit feedforwarded operation, see quantum circuits in Fig. 5a. In Kitaev’s phase estimation and orderfinding algorithms, the computational capacity is determined by the number of nqubits in the yregister, and the computational accuracy is determined by the number of msequences in the xregister. In this respect, one can see processing quantum algorithms with qudits results in nontrivial advantages: a log_{2}(d) larger computational capacity, and log_{2}(d) higher computational accuracy or log_{2}(d)less computational steps to achieving the same precision, as shown in Supplementary Fig. 3b, which could be important to quantum computers with limited coherence time.
In the quantum phase estimation, we aim to compute the eigenphase ϕ of an unitary as \(O\left\psi \right\rangle \)=\({e}^{{\mathbb{i}}2\pi \phi }\left\psi \right\rangle \), given the eigenstate of \(\left\psi \right\rangle \). The eigenphase of ϕ can be described in dary as 0. ϕ_{1}ϕ_{2}…ϕ_{m−1}ϕ_{m}, where m denotes iterative steps determining the approximation accuracy^{26,29}, and each dit of the phase is in [0, 1,..., d − 1]^{67,69,73}. We take the sth step as an example (see quantum circuit in Fig. 5a). We prepare an input state of \(\frac{1}{\sqrt{d}}\mathop{\sum }\nolimits_{j = 0}^{d1}\leftj\right\rangle \left\psi \right\rangle \) and perform the MVCU gate, that results in a state of \(\frac{1}{\sqrt{d}}\mathop{\sum }\nolimits_{j = 0}^{d1}{e}^{{\mathbb{i}}j2\pi (0.{\phi }_{s}{\phi }_{s+1}...{\phi }_{m})}\leftj\right\rangle \left\psi\right\rangle \). Then, the xregister qudit state is feedforwardly rotated around the Pauli Z_{d} basis as diag[\(1,{e}^{{\mathbb{i}}2\pi {\theta }_{s}},...,{e}^{{\mathbb{i}}2\pi (d1){\theta }_{s}}\)], where the rotation angle θ_{s} of − 0.0ϕ_{s+1}ϕ_{s+2}…ϕ_{m} is given by previous measurement outcomes. Remarkably, implementing an inverse F_{d} in the xregister returns an output state as \({\left{\phi }_{s}\right\rangle }_{}=\lefts\right\rangle \) (see Supplementary Note 5). Measuring the xregister in the computational basis of \(\lefts\right\rangle \) therefore allows the extraction of the sth dit of the dit expansion. The algorithm iteratively computes all m dits of the eigenphase backwardly, in which, notably, each dit is once estimated with dary accuracy. Figure 5b–d report measured eigenphases of 4dimensional unitary matrices by quaternary phase estimation. We estimated the four eigenphases for three logic gates, i.e., a phase gate Z_{4}, a Fourier gate F_{4} and a randomised gate U_{random} (see their explicit forms in Supplementary). Each pie chart presents one dit measurement outcomes, and the area of each coloured sector denotes measured probability distributions in the computational basis of {\(\left0\right\rangle \),\(\left1\right\rangle \),\(\left2\right\rangle \),\(\left3\right\rangle \)}, respectively. In Fig. 5b, c, the eigenphases of Z_{4} and F_{4} gates can be described by a single dit. Figure 5d shows the computed eigenphases of the U_{random} gate with an accuracy of 12 dits, by running the algorithm with a number of 12 interactions on the dQPU. Instead, in the qubitbased device, achieving the same computational accuracy of ± 4^{−12} requires a number of 24 computational interactions. And the achieved computational accuracies of 12 quarts are sufficient for the calculation of molecular eigenenergies^{67,69}. In Fig. 5, it shows experimental data are in good agreement with theoretical predictions (indicated under each pie).
The task of quantum factoring is to efficiently compute the prime factors p and q from an integer N^{25}. It can be reduced to the task of finding the order r of a module N, i.e., by computing a function f: a^{r}modN = 1 (a is a coprime of N), and with a high probability it returns a factor as gcd(a^{r/2} ± 1, N), where gcd(α, β) refers to the greatest common divisor of α and β. As the orderfinding is just the phase estimation of a unitary having the eigenphases of s/r, s ∈ [0, r − 1], one can directly adopt the dary phase estimation to determine the order of r in the dary format. It can be considered as a generalisation of dary orderfinding by adopting Kiteav’s iterative techniques^{70,71,72} (see details in Supplementary Note 6). We then reprogrammed the dQPU to implement the orderfinding in quaternary. The a ∈ [0, r − 1] satisfying gcd(a, 15) = 1 is randomly chosen. We chose a = 4 and 2 as examples, and set the unitary of the MVCU gate as {I_{d}, X_{d}, I_{d}, X_{d}} and \(\{{I}_{d},{X}_{d},{X}_{d}^{2},{X}_{d}^{3}\}\), respectively, where I_{d} is the dmode identity. In our experiment, the orderfinding algorithm was iteratively implemented by three steps, and each step returns quaternary outcomes in the computational basis, resulting in the 3quart (64level) computational accuracy of the s/r eigenphase. Figure 5e, f show the measured output probabilities of the xregister in the computational basis of \(\leftijk\right\rangle \), i, j, k = 0, 1, 2, 3. Classical statistic fidelities F_{c} of 0.909(9) and 0.922(9) were obtained in comparison with theoretical distributions, showing successful estimations of the order of r = 2 (Fig. 5e) and r = 4 (Fig. 5f), respectively. The dQPU thus finds the order with doubleenhanced computational accuracy; alternatively speaking, it executes the task twice faster than a qubitQPU, given the same estimation precision. The orderfinding algorithm together with classical processing using the continued fraction algorithm returns the factor of gcd(a^{r/2} ± 1, N) = (3, 5). Implementing the dary algorithms in the dQPU can therefore find the order of a function and compute the eigenphase of a unitary, with a log_{2}(d)faster computational speed.
Discussion
We have reported a proofofprinciple experimental demonstration of a programmable quditbased quantum processor in photonic integrated circuits, and implementations of several generalised dary quantum Fourier transform algorithms in the dQPU chip. In agreement with the references^{17,18,19,24,43,44,45,46}, our experimental results show that quditbased quantum computation with integrated photonics can enhance quantum parallelism in terms of the computational capacity, accuracy and efficiency, in comparison with its qubitbased quantum computing counterpart. The computational capacity of the two ququart quantum processor is equivalent to that of a fourqubit processor, thus allowing the implementations of the Deutsch’s algorithms for a function with longerstring. Keeping the same number of photons n but encoding each qudit in a dimension d, not only gives a larger Hilbert space^{74}, but also significantly improve the detection rate of photons^{43,44}. We obtained the detection rate of about 6 orders brighter than that in another device with the same Hilbert space^{58}. More analysis is provided in Supplementary Fig. 3. Moreover, multiple parallel evaluations of the function and multiple path interference in the dary quantum Fourier gate, allow the enhancement of the computational efficiency and speed up of the determination of desired solutions. In the implementations of Kitaev’s phase estimation and factorisation, a number of log_{2}(d)less iterations are needed in the qudit processor, i.e., a log_{2}(d) times speed up of quantum computation, compared with the qubit ones, given the same computational accuracy (see Supplementary Fig. 3b).
As the multivalue quantum controlled gates are the result of the entanglement in the generation stage and the gates are instead local operations that steer the state to collapse in the desired outputs, our scheme can be straightforwardly generalised to multiqudit quantum computiation. Its scalability is naively dependent on the number (n) of qudits and the dimensionality (d) of each qudit. Regarding the dimension of units, though the ququart states are implemented as an example in this work, it is straightforward to extend to a largerd device^{36}, which can be fabricated using the same CMOS fabrication techniques. Remarkably, this entanglementassisted dQPU scheme works with a success probability of 1/d regardless of n (Supplementary Note 3). The scaling of dQPU therefore strongly relies on the generation of the qudit GHZ entangled states. Combing the stateoftheart technologies, including the techniques of generating multiphoton qudit GHZ states^{45,46}, onchip highfidelity control of qudit states^{36}, highquality photonpair sources^{75,76}, lowloss fibrechip interface^{75,77}, and largescale quantum integration^{57}, we estimate a 10photons dQPU is achievable in near term. Its further scaling requires highefficiency heralded multiplexing photonpairs sources^{78} and multiplexing qudit GHZ generators^{31}. That being said, given the efficient generation of the multiphoton multiqudit GHZ states, the dQPU scheme is scalable. Calculations and analysis are provided in Supplementary Note 9 and Supplementary Fig. 3d. Moreover, when scaling up the dQPU, an interesting concern is the required resources, in particular the number of classical controls. As an example, let us consider a processor with one qudit in the auxiliary register and n qudits in the data register (see Fig. 2a). It requires a number of (n + 1) singlequdit generators for state preparation, (nd) local singlequdit operators for multiqudit MVCU operation, and (dn + 1) singlequdit projectors. The physical resources, i.e, the number of phaseshifters, scale with (d^{2} − d) for the qudit operators^{50,59}, and 2(d − 1) for the qudit generators and projectors^{36}, as shown in Supplementary Note 9 and Supplementary Fig. 2. Importantly, the required resources for classical controls scale polynomially with the number of particles. In Supplementary Fig. 3c, it is shown that thousands of phaseshifters are required for a 10photon dQPU. This large amount of phaseshifters can be individually addressed and controlled, by using a cointegration technology of photonic and electronic circuits in silicon.
The highly flexible and reliable programmability of the qudit processor, that is enabled by technological advances in a monolithic integration of all key functionalities and capabilities in a silicon chip, has allowed the implementations of more than one million qudit generators, operators and projectors (see Supplementary Note 8), and also the benchmarking of different generalised quantum algorithms. Such programmability can transition the advanced technologies in controlling qudit states and gates^{36,37,38,39,40,41,42,43,44,45,46,47,48} to algorithm implementations, playing an enabling role in the roadmap of quditbased quantum computations. The full chipscale integration technologies also perfectly match the topdown hierarchy of quantum computing, in which users can define and execute multiple quantum tasks by recompiling the software and reprogramming the quantum hardware. In general, the programmable quditbased quantum devices can find applications in noiseresilient quantum network^{9,10}, quantum simulation of complex chemical and physical systems^{12,13,14,15}, and universal quantum computing with qudit cluster states^{19,20,21}.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code availability
The codes used for the analysis included in the current study are available from the corresponding authors upon reasonable request.
References
Giustina, M. et al. Significantloopholefree test of bell’s theorem with entangled photons. Phys. Rev. Lett. 115, 250401 (2015).
Shalm, L. K. et al. Strong loopholefree test of local realism. Phys. Rev. Lett. 115, 250402 (2015).
Liao, S. K. et al. Satellitetoground quantum key distribution. Nature 549, 43–47 (2017).
Liao, S. K. et al. Satelliterelayed intercontinental quantum network. Phys. Rev. Lett. 120, 030501 (2018).
Arute, F. et al. Quantum supremacy using a programmable superconducting processor. Nature 574, 505–510 (2019).
Wu, Y. et al. Strong quantum computational advantage using a superconducting quantum processor. Phys. Rev. Lett. 127, 180501 (2021).
Collins, D. et al. Bell inequalities for arbitrarily highdimensional systems. Phys. Rev. Lett. 88, 040404 (2002).
Vértesi, T., Pironio, S. & Brunner, N. Closing the detection loophole in bell experiments using qudits. Phys. Rev. Lett. 104, 060401 (2010).
Cerf, N. J. et al. Security of quantum key distribution using dlevel systems. Phys. Rev. Lett. 88, 127902 (2002).
Bouchard, F. et al. Highdimensional quantum cloning and applications to quantum hacking. Sci. Adv. 3, e1601915 (2017).
Islam, N. T. et al. Provably secure and highrate quantum key distribution with timebin qudits. Sci. Adv. 3, e1701491 (2017).
Kaltenbaek, R. et al. Optical oneway quantum computing with a simulated valencebond solid. Nat. Phys. 6, 850–854 (2010).
Neeley, M. et al. Emulation of a quantum spin with a superconducting phase qudit. Science 325, 722–725 (2009).
Blok, M. S. et al. Quantum information scrambling on a superconducting qutrit processor. Phys. Rev. X. 11, 021010 (2021).
Choi, S. et al. Observation of discrete timecrystalline order in a disordered dipolar manybody system. Nature 543, 221–225 (2017).
Wei, S. & Long, G. Duality quantum computer and the efficient quantum simulations. Quant. Inf. Process. 15, 1189–1212 (2016).
Luo, M. & Wang, X. Universal quantum computation with qudits. Sci. China Phys. Mech. 57, 1712–1717 (2014).
Zhou, D. L. et al. Quantum computation based on dlevel cluster state. Phys. Rev. A. 68, 062303 (2003).
Wei, T. C., Affleck, I. & Raussendorf, R. Affleckkennedyliebtasaki state on a honeycomb lattice is a universal quantum computational resource. Phys. Rev. Lett. 106, 070501 (2011).
Campbell, E. T. Enhanced faulttolerant quantum computing in dlevel systems. Phys. Rev. Lett. 113, 230501 (2014).
Bocharov, A., Roetteler, M. & Svore, K. M. Factoring with qutrits: Shor’s algorithm on ternary and metaplectic quantum architectures. Phys. Rev. A. 96, 012306 (2017).
Gokhale, P. et al. Asymptotic improvements to quantum circuits via qutrits. In Proceedings of the 46th International Symposium on Computer Architecture, 554566 (2019).
Wang, D. S., Stephen, D. T. & Raussendorf, R. Qudit quantum computation on matrix product states with global symmetry. Phys. Rev. A. 95, 032312 (2017).
Wang, Y. et al. Qudits and highdimensional quantum computing. Front. Phys. 8, 479 (2020).
Shor, P. W. Algorithms for quantum computation: Discrete logarithms and factoring. Proc. 35nd Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press (1994).
Kitaev., A. Y. Quantum measurements and the abelian stabilizer problem. Electronic Colloquium on Computational Complexity. 3 (1996).
Griffiths, R. B. & Niu, C. S. Semiclassical fourier transform for quantum computation. Phys. Rev. Lett. 76, 3228–3231 (1996).
Parker, S. & Plenio, M. B. Efficient factorization with a single pure qubit and \(\log N\) mixed qubits. Phys. Rev. Lett. 85, 3049–3052 (2000).
Dobšíček, M. et al. Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A twoqubit benchmark. Phys. Rev. A. 76, 030306 (2007).
Adcock, M. R. A., Høyer, P. & Sanders, B. C. Quantum computation with coherent spin states and the close hadamard problem. Quant. Inf. Process. 15, 1361–1386 (2016).
Paesani, S. et al. Scheme for universal highdimensional quantum computation with linear optics. Phys. Rev. Lett. 126, 230504 (2021).
Erhard, M., Krenn, M. & Zeilinger, A. Advances in highdimensional quantum entanglement. Nat. Rev. Phys. 2, 365–381 (2020).
Ringbauer, M. et al. A universal qudit quantum processor with trapped ions. arXiv:2109.06903 [quantph] (2021).
CerveraLierta, A. et al. Experimental highdimensional greenbergerhornezeilinger entanglement with superconducting transmon qutrits. arXiv:2104.05627 [quantph] (2021).
Chen, X. et al. Quantum entanglement on photonic chips: a review. Adv. Photon. 3, 064002 (2021).
Wang, J. et al. Multidimensional quantum entanglement with largescale integrated optics. Science 360, 285–291 (2018).
Li, L. et al. Metalensarray–based highdimensional and multiphoton quantum source. Science 368, 1487–1490 (2020).
Kues, M. et al. Quantum optical microcombs. Nat. Photon. 13, 170–179 (2019).
Hu, X. M. et al. Efficient generation of highdimensional entanglement through multipath downconversion. Phys. Rev. Lett. 125, 090503 (2020).
Dada, A. C. et al. Experimental highdimensional twophoton entanglement and violations of generalized Bell inequalities. Nat. Phys. 7, 677–680 (2011).
Feng, L. F. et al. Onchip coherent conversion of photonic quantum entanglement between different degrees of freedom. Nat. Commun. 7, 11985 (2016).
Martin, A. et al. Quantifying photonic highdimensional entanglement. Phys. Rev. Lett. 118, 110501 (2017).
Reimer, C. et al. Highdimensional oneway quantum processing implemented on dlevel cluster states. Nat. Phys. 15, 148–153 (2019).
Imany, P. et al. Highdimensional optical quantum logic in large operational spaces. npj Quant. Inf. 5, 59 (2019).
Malik, M. et al. Multiphoton entanglement in high dimensions. Nat. Photon. 10, 248–252 (2016).
Erhard, M. et al. Experimental greenberger–horne–zeilinger entanglement beyond qubits. Nat. Photon. 12, 759–764 (2018).
Hu, X. M. et al. Experimental highdimensional quantum teleportation. Phys. Rev. Lett. 125, 230501 (2020).
Luo, Y. H. et al. Quantum teleportation in high dimensions. Phys. Rev. Lett. 123, 070505 (2019).
Zhong, H. S. et al. Phaseprogrammable gaussian boson sampling using stimulated squeezed light. Phys. Rev. Lett. 127, 180502 (2021).
Carolan, J. et al. Universal linear optics. Science 349, 711–716 (2015).
Debnath, S. et al. Demonstration of a small programmable quantum computer with atomic qubits. Nature 536, 63–66 (2016).
Monroe, C. et al. Programmable quantum simulations of spin systems with trapped ions. Rev. Mod. Phys. 93, 025001 (2021).
Watson, T. F. et al. A programmable twoqubit quantum processor in silicon. Nature 555, 633–637 (2018).
Wang, J. et al. Integrated photonic quantum technologies. Nat. Photon. 14, 273–284 (2020).
Pelucchi, E. et al. The potential and global outlook of integrated photonics for quantum technologies. Nat. Rev. Phys. https://doi.org/10.1038/s4225402100398z (2021).
Wang, J. et al. Experimental quantum Hamiltonian learning. Nat. Phys. 13, 551–555 (2017).
Chen, X. et al. A generalized multipath delayedchoice experiment on a largescale quantum nanophotonic chip. Nat. Commun. 12, 2712 (2021).
Adcock, J. C. et al. Programmable fourphoton graph states on a silicon chip. Nat. Commun. 10, 3528 (2019).
Clements, W. R. et al. Optimal design for universal multiport interferometers. Optica 3, 1460–1465 (2016).
Riofrío, C. A. et al. Experimental quantum compressed sensing for a sevenqubit system. Nat. Commun. 8, 15305 (2017).
Mohseni, M., Rezakhani, A. T. & Lidar, D. A. Quantumprocess tomography: resource analysis of different strategies. Phys. Rev. A. 77, 032322 (2008).
Hofmann, H. F. Complementary classical fidelities as an efficient criterion for the evaluation of experimentally realized quantum operations. Phys. Rev. Lett. 94, 160504 (2005).
Fan, Y. A generalization of the deutschjozsa algorithm to multivalued quantum logic. In 37th International Symposium on MultipleValued Logic, 12–12 (IEEE, 2007).
Deutsch, D. & Jozsa, R. Rapid solution of problems by quantum computation. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 439, 553–558 (1992).
Bernstein, E. & Vazirani, U. Quantum complexity theory. SIAM J. Comput. 26, 1411–1473 (1997).
AspuruGuzik, A. et al. Simulated quantum computation of molecular energies. Science 309, 1704–1707 (2005).
Lanyon, B. P. et al. Towards quantum chemistry on a quantum computer. Nat. Chem. 2, 106–111 (2010).
Zhou, X. Q. et al. Calculating unknown eigenvalues with a quantum algorithm. Nat. Photon. 7, 223–228 (2013).
Santagati, R. et al. Witnessing eigenstates for quantum simulation of hamiltonian spectra. Sci. Adv. 4, aap9646 (2018).
Smolin, J. A., Smith, G. & Vargo, A. Oversimplifying quantum factoring. Nature 499, 163–165 (2013).
MartinLopez, E. et al. Experimental realization of shor’s quantum factoring algorithm using qubit recycling. Nat. Photon. 6, 773 (2012).
Monz, T. et al. Realization of a scalable Shor algorithm. Science 351, 1068–1070 (2016).
Lu, H. H. et al. Quantum phase estimation with timefrequency qudits in a single photon. Adv. Quant. Technol. 3, 1900074 (2020).
Vigliar, C. et al. Errorprotected qubits in a silicon photonic chip. Nat. Phys. 17, 1137–1143 (2021).
Llewellyn, D. et al. Chiptochip quantum teleportation and multiphoton entanglement in silicon. Nat. Phys. 16, 148–153 (2020).
Paesani, S. et al. Nearideal spontaneous photon sources in silicon quantum photonics. Nat. Commun. 11, 2505 (2020).
Paesani, S. et al. Generation and sampling of quantum states of light in a silicon chip. Nat. Phys. 15, 925–929 (2019).
Kaneda, F. & Kwiat, P. G. Highefficiency singlephoton generation via largescale active time multiplexing. Sci. Adv. 5, eaaw8586 (2019).
Acknowledgements
We acknowledge X.Wang and S.Tao for useful discussions and assistance of experiment. We acknowledge support from Beijing Natural Science Foundation (Z190005), the National Key R&D Program of China (nos 2019YFA0308702, 2018YFB1107205, 2018YFB2200403, and 2018YFA0704404), the National Natural Science Foundation of China (nos 61975001, 61590933, 61904196, 61675007, and 61775003), and Key R&D Program of Guangdong Province (2018B030329001).
Author information
Authors and Affiliations
Contributions
J.W. conceived the project. Y.C., J.H., Z.C.Z., J.M., Z.N.Z, X.C., C.Z., J.B., T.D., H.Y., M.Z., B.T., and Y.Y. implemented the experiment. Y.C., X.C., and J.B. designed the device. Y.C., J.H., Z.C.Z., J.M. and Z.N.Z provided theoretical analysis. D.D., Z.L., Y.D., L.K.O., M.G.T., J.L.O., Y.L., Q.G., and J.W. managed the project. All authors discussed the results and contributed to the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks Jacqui Romero and Taira Giordani for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chi, Y., Huang, J., Zhang, Z. et al. A programmable quditbased quantum processor. Nat Commun 13, 1166 (2022). https://doi.org/10.1038/s4146702228767x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4146702228767x
This article is cited by

An elementary review on basic principles and developments of qubits for quantum computing
Nano Convergence (2024)

Demonstration of hypergraphstate quantum information processing
Nature Communications (2024)

Monolithic backendofline integration of phase change materials into foundrymanufactured silicon photonics
Nature Communications (2024)

Navigating the 16dimensional Hilbert space of a highspin donor qudit with electric and magnetic fields
Nature Communications (2024)

Quantum manybody simulations on digital quantum computers: Stateoftheart and future challenges
Nature Communications (2024)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.