Abstract
Quantum Walks are unitary processes describing the evolution of an initially localized wavefunction on a lattice potential. The complexity of the dynamics increases significantly when several indistinguishable quantum walkers propagate on the same lattice simultaneously, as these develop nontrivial spatial correlations that depend on the particle’s quantum statistics, mutual interactions, initial positions, and the lattice potential. We show that even in the simplest case of a quantum walk on a one dimensional graph, these correlations can be shaped to yield a complete set of compact quantum logic operations. We provide detailed recipes for implementing quantum logic on onedimensional quantum walks in two general cases. For noninteracting bosons—such as photons in waveguide lattices—we find highfidelity probabilistic quantum gates that could be integrated into linear optics quantum computation schemes. For interacting quantumwalkers on a onedimensional lattice—a situation that has recently been demonstrated using ultracold atoms—we find deterministic logic operations that are universal for quantum information processing. The suggested implementation requires minimal resources and a level of control that is within reach using recently demonstrated techniques. Further work is required to address errorcorrection.
Introduction
Quantum walks (QWs) are unitary processes describing the propagation of quantum particles on lattice potentials.^{1,2,3} Originally described as a quantummechanical analog of the classical random walk, QWs were found to exhibit faster propagation and enhanced sensitivity to lattice parameters due to their coherent nature.^{3} These properties generated broad interest in applying QWs to quantum information processing tasks.^{4}
Experimentally, QWs have been implemented using a wide array of platforms, including photonics,^{5,6,7,8,9,10,11,12} trapped ions,^{13,14} and ultracold atoms.^{15,16,17} The current degree of experimental control of these systems is remarkable: it is possible to prepare an initial state with singlesite and singleparticle resolution, to control almost every aspect of the lattice potential, and to directly monitor the evolving wave function. Early experiments demonstrated the behavior of singleparticle QWs; however, these dynamics can be desribed by classical wave equations (indeed, some of these experiments were performed with coherent light)^{5,6,7,9}), and thus cannot display nonclassical features. Nonclassical behavior can be observed when several indistinguishable particles participate in the QW simultaneously, as was shown both theoretically^{7,12,18,19,20} and experimentally.^{8,21} Here, nonclassical spatial correlations—i.e., nontrivial dependencies between the positions of different walkers—emerge due to quantum (bosonic or fermionic) statistics.^{7,19,22} Recent work has investigated the role of interactions in the twoparticle quantum walk, finding that they give rise to even more complex correlated dynamics.^{23,24} These ‘strongly correlated’ QWs were recently observed experimentally in a system of ultracold atoms.^{17}
Can the spatial correlations that emerge between several quantum cowalkers be useful for quantum information processing? Recent theoretical work by Childs et. al. demonstrated that, in principle, multiparticle QWs could be used to implement universal quantum computation.^{25} However, the geometry and complexity of the required lattice potential in the proposed scheme are far beyond current experimental capabilities.
In this work, we show how controlling the lattice potential of a QW can impose certain spatial correlations between walkers. Using this approach, we design quantum logic gates on a simple onedimensional array of potential wells using minimal resources: one quantum walker and a small number of lattice sites per qubit. For noninteracting bosons (such as photons in waveguide lattices) we find that this approach yields highfidelity probabilistic logic gates with a similar success rate to those found previously—but with a much simpler physical design. For interacting bosonic quantum walkers (e.g., ultracold atoms in optical potentials or photons in nonlinear devices) we find that a complete set of highfidelity quantum logic gates can be realized using a linear array of potential wells with nearestneighbor coupling and only two sites per qubit, demonstrating the universality of this architecture for quantum computation.
While our analysis is general to any system that can support quantum walk dynamics, here we focus on two physical systems in which our results can be implemented using existing experimental capabilities: noninteracting, indistinguishable photons in waveguide lattices and interacting ultracold bosonic atoms trapped in an optical lattice. In the tightbinding limit, both systems can be described by the same timeindependent BoseHubbard equation:^{7,23}
where in the atomic (photonic) case, E_{m} is the onsite energy (index of refraction) of site m, \(a_m^\dagger {\mathrm{\backslash }}a_m\) is the creation/annihilation operator for an atom (photon) at site m, \(\hat n_m = a_m^\dagger a_m\) is the number operator, and J_{l,m} ≤ 0 is the tunneling rate between nearest sites. Finally, Γ is the onsite interaction energy, i.e., an energy cost for the occupancy of two or more bosons on the same site; in the linear photonic case discussed below, Γ = 0, while in the atomic case this value is usually nonzero and can be adjusted experimentally. While the atomic system evolves in time according to the unitary operator U^{(atomic)} = e^{−iHt}, the photons also evolve in space along the z direction U^{(photonic)} = e^{−iHz} (see Fig. 1d).
Results
Defining qubits on a lattice
The continuoustime QW is described by the unitary evolution under the Hamiltonian described in Eq. (1).^{2,6,7,23} However, the basic element of interest for quantum gates of the type discussed here is the quantum bit or qubit. To define our qubits on the lattice, we use a socalled dualrail encoding where a qubit is physically implemented by a single boson in a pair of neighboring potential wells (see Fig. 1), with the states \(\left 0 \right\rangle\) and \(\left 1 \right\rangle\) of the qubit defined by the particle being in the left or right well. A single quantum particle can occupy the two sites in a superposition, encoding a qubit without the need for additional degrees of freedom. In this way, a system of n qubits can be realized in one dimension using n bosons and N = 2n lattice sites, with one boson in the first two sites (representing the first qubit), one boson in the next two sites (representing the second qubit), and so forth. As we discuss below, in some cases the implementation requires additional auxiliary sites (typically one per qubit). Note that in this geometry, many physically permitted lattice states (e.g., those with more than one particle on the same site) are not members of the logic space (i.e., the multiqubit tensorproduct space). Nevertheless, we show that it is possible to engineer the lattice parameters such that, at time t = t_{ final } = 1, the transformation U = e^{−iH} maps logic states to other logic states with high fidelity.
Implementing quantum gates
Having defined our qubits, we turn to the task of designing a universal set of quantum gates, i.e., finding lattice parameters that yield desired unitary transformations on the logical space. Designing and building quantum logic gates remains one of the most difficult aspects of quantum computing, and our case is no exception. The reason for this difficulty is simple: from the physical description of a given device—in our case, the lattice parameters—it is straightforward to write down the manyparticle Hamiltonian H and from it to calculate the unitary evolution operator U = e^{−iH} that fully describes the operation of the device. The inverse problem, however, is hard: given a desired multiparticle unitary U, it is difficult to find a corresponding Hamiltonian that meets the physical and geometrical constraints of the device, e.g., the onedimensionality of the lattice. Beyond these restrictions, the primary difficulty from a theoretical perspective is the lack of an analytic inverse to the matrix exponential or to the transformation function from singleparticle behavior to multiparticle behavior. Even for a given unitary U there are an infinite number of Hermitian matrices H satisfying e^{−iH} = U; it is unclear how to determine constructively which of these satisfies the other system constraints. Furthermore, if the logical quantum states are only a subset of the full Hilbert space, then the quantum gate operation is only a submatrix of the overall evolution operator U. In this case, U is not even uniquely defined by the desired gate operation. As described below, we tackle these difficulties using a combined analytical and computational approach that finds appropriate lattice parameters to achieve a given ideal gate operation with high fidelity.
There are many options for the choice of a universal set of gates. One useful choice is the gate set of the controlledNOT (CNOT) operation, along with either all singlequbit rotations (exactly universal) or the Hadamard and phaseshift singlequbit gates (approximately universal).^{26} We first discuss the singlequbit gates, which are straightforward to calculate analytically. These gates are applicable to both interacting and noninteracting systems. We then elaborate on the construction of the CNOT gate.
Singlequbit gates
We present the exact construction for a set of singlequbit gates. Since these are onequbit operations, they are implemented using a single particle in two lattice sites. As such, the interaction (Γ) is irrelevant and the matrix of lattice parameters
$$G = \left( {\begin{array}{*{20}{c}} {E_1} & {J_{12}} \\ {J_{12}} & {E_2} \end{array}} \right)$$(with J_{12} ≤ 0) can be directly interpreted as the singleparticle Hamiltonian. The unitary gate, obtained by evolving with G for a time t_{ final } = 1, is then U = e^{−iG}.
One simple universal quantum gate set includes the Hadamard gate, the phaseshift gate, and the CNOT gate.^{26} The phaseshift gate is the simplest to implement. It is composed of two decoupled lattice sites in which the onsite energy between the sites is detuned. Specifically, to implement the singlequbit phaseshift operator \(R_\theta = \left( {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & {e^{i\theta }} \end{array}} \right).\), one may apply the singlequbit Hamiltonian
Next in complexity is the singlequbit Hadamard gate \(H = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} 1 & 1 \\ 1 & {  1} \end{array}} \right)\). The Hamiltonian and propagation time that generates the Hadamard transformation can be solved analytically, and is given by
Note that a simple tunneling between two identical wells for half the tunneling time is not exactly analogous to the operation of a beamsplitter in linear optics, as it does not reproduce the Hadamard gate. Under our Hamiltonian dynamics the splitting is symmetric in phase and therefore modified tunneling rates and additional diagonal terms are required to adjust the output phases. As a result, the operation of the CNOT gate constructed using integrated waveguide beamsplitters^{27} is not identical to previous implementations that used the same design in bulk optics.^{28} The difference between these implementations becomes apparent when comparing the complex values of the unitary operations rater then the transition probabilities.
An alternative (exactly) universal gate set includes the CNOT together with all singlequbit unitaries. Any singlequbit unitary, U, can be implemented by first decomposing it in the form^{26}
$$U = e^{i\alpha }R_z(\beta )R_x(\gamma )R_z(\delta ) = e^{i\alpha }R_z(\beta )HR_z(\gamma )HR_z(\delta )$$where the zrotation \(R_z(\theta ) = \left( {\begin{array}{*{20}{c}} {e^{  i\theta /2}} & 0 \\ 0 & {e^{i\theta /2}} \end{array}} \right)\) can be implemented with the Hamiltonian
$$G_{R_z(\theta )} = \frac{\theta }{2}\left( {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & {  1} \end{array}} \right){\kern 1pt} ,$$and the xrotation \(R_x(\theta ) = {\mathrm{exp}}\left( {\begin{array}{*{20}{c}} 0 & {  i\theta {\mathrm{/}}2} \\ {  i\theta {\mathrm{/}}2} & 0 \end{array}} \right)\) can be implemented either with the Hamiltonian
or by conjugating R_{ z }(θ) by the Hadamard operation H described earlier. The phase e^{iα} can be implemented with \(G_\alpha =  \alpha \left( {\begin{array}{*{20}{c}} 1 & 0 \\ 0 & 1 \end{array}} \right)\).
CNOT gate using interacting QWs
We first consider the case in which the quantumwalking particles interact when occupying the same lattice site, i.e., Γ ≠ 0. To design the twoqubit CNOT gate using the dualrail encoding, we consider a lattice with four sites and two bosons. This problem then is defined by eight lattice parameters: four onsite potential terms (E_{ m } in Eq. (1)), three tunneling terms (J_{l,m}), and the interaction parameter (Γ). The complete twobody Hamiltonian H is described by a 10 × 10 matrix (the size of the Hilbert space for two bosons in four modes). To perform the logical gate operation, the system evolves according to U = e^{−iH}. The CNOT gate operation is then given by a 4 × 4 submatrix of U over the logical states \(\left {1010} \right\rangle\), \(\left {1001} \right\rangle\), \(\left {0110} \right\rangle\), \(\left {0101} \right\rangle\) (presented here in the photon occupationnumber basis); the other six basis states, while physically allowed, are not members of the logical basis.
As mentioned above, finding the physical lattice parameters from the desired gate is a nontrivial inverse problem. Using nonlinear optimization techniques^{29,30,31,32} (detailed in the methods section), we optimized the eight parameters of the system to maximize the fidelity of the gate when acting on the logical input states under the constraints that the parameters represent a physical onedimensional lattice, meaning that the onsite parameters are real, that the tunneling parameters are real, nonpositive, and connect only nearestneighboring sites, and that the values of the onsite, tunneling, and interaction terms are within experimentally relevant bounds. Specifically, we demand that 0 ≥ J_{l,m} ≥ −J_{max}, −J_{max} ≥ E_{ m } ≥ J_{max}, and Γ ≤ Γ_{max}, where J_{max} and Γ_{max} are the largest allowed tunneling rate and interaction level in the optimization protocol. In our optimization we set J_{max} = 4π, limiting the maximal number of tunneling events (or Rabioscillations) to 4. In practice, this experimental bound is dictated by the loss and decoherence rate of the system, determining the maximal relevant propagation time. We also set Γ_{max} = 10J_{max}.
An example of a resulting lattice that yields the twoqubit CNOT gate is given (to two decimal places) by
with interaction strength Γ = 21.68π. Here, the diagonal and offdiagonal entries of G_{CNOT} represent the parameters E_{ m } and J_{l,m}, respectively, of the Hamiltonian H. Eq. (5) represents a recipe for a foursite lattice that yields a CNOT gate with fidelity (as defined in the methods section) of 99.6%. This gate’s operation is summarized in Fig. 2.
If the bounds on the parameters are relaxed, the fidelity moves even closer to 100%. Figure 3 summarizes the optimization results. Figure 3a shows the convergence of independent runs with random starting points to the same final result. Figure 3b presents the expected gate fidelity vs. the maximally allowed values of the interaction Γ_{max}. For a fixed maximal tunneling of J_{max} = 4π, the fidelity achieves a value close to 0.95 at Γ_{max}/J_{max} = 0.5 and then slowly approaches unity as this value is further increased. In a system with a given Γ_{max}, it is still possible to improve the fidelity further by increasing the rate of the coupling between sites, i.e., increasing J_{max}; see Fig. 3c.
Based on a basic analysis of the effect of imperfections and noise, we found that within the experimental parameters, these are expected to have a negligible effect on the fidelity of the CNOT gate. These results are presented in the Supplementary Information. We therefore expect the main reduction in fidelity to occur between the application of sequential gates, when the parameters of the system are modified—encouragingly, an experimental demonstration of highfidelity switching between potentials was reported recently.^{33}
CNOT gate for noninteracting QW
A related procedure can be derived to generate a new type of QWbased photonic logic gates. The use of single photons in linearoptical setups for implementing quantum gates is a fastdeveloping branch of quantum information science.^{28,34} As photons do not easily interact, the devices are usually linear, and as a result can perform quantum operations only probabilistically, i.e., with a nonzero probability of having an output that does not correspond to a valid logical state. However, as shown by Knill et al.,^{35} by combining these probabilistic gates with effective nonlinearities induced by measurement, it is in principle possible to build a quantum computation device that scales efficiently.
In linear optics, the entries of the multiparticle unitary can be calculated as functions of the singleparticle unitary that describes the evolution of one particle in the device.^{20} Given this singleparticle unitary, Reck et al. showed how to configure a network of beamsplitters and phaseshifters to implement the unitary physically. Following this finding, current quantum photonic gates—whether using bulk or integrated optics—are usually based on the beamsplitter architecture. For example, Ralph et al.^{36} proposed a probabilistic CNOT gate for photons with a success probability of 1/9, based on the beamsplitter architecture. This gate was successfully implemented using bulk optics.^{28} Later, the same design was reimplemented in integrated photonics,^{27} where a different hardware (waveguide couplers) was used to implement the same beamsplitter arrangement on a chip. However, in the case of integrated photonics, mimicking the beamsplitter architecture is not obviously the best choice: it utilizes only pairwise waveguide couplings, and it introduces bends to the waveguide design, and as the bends cannot be too sharp, this results in longer devices with increased losses.
In this section we show how to implement the quantum CNOT gave using the QW of photons on a small lattice. In photonics, continuoustime QWs are implemented on a waveguide lattice in which all the waveguides are straight and laid out in parallel (see Fig. 1d).^{6,7} Photons are injected into lattice sites and hop between the waveguides as they propagate along the lattice, according to Eq. (1). Onedimensional photonic walks are especially versatile, as this approach for integrated photonic devices allows for various additional components to be added, such as integrated sources, detectors, and modulators.
Since we expect that only a probabilistic CNOT gate is attainable, we allow for vacuum ancilla lattice sites. Thus, our Hamiltonian H defines a onedimensional QW of 2 particles in n sites, and is described by n onsite terms and n − 1 coupling coefficients, where n may exceed the four sites defining two qubits. The multiparticle unitary evolution operator U = e^{−iHT} is an \(\frac{{n(n + 1)}}{2} \times \frac{{n(n + 1)}}{2}\) matrix that, as before, contains the CNOT gate as a 4 × 4 submatrix.
Using both global and local nonlinear optimization procedures (see Methods), we optimized the gate fidelity and success probability over a space of lattice parameters, restricted to physically reasonable values, that would yield the correct manyparticle operation. The results of this optimization for the CNOT gate are presented in Fig. 4. The curve shows the maximum fidelity of the obtained gate operation as a function of required success probability. Here our lattice contains six sites—the 4 sites required for two qubits, as well as two auxiliary sites—as sketched in Fig. 1d. A 100.00% fidelity gate can be found for a success probability of 1/9 (as in the beamsplitter approach), with the lattice (to two decimal places):
Again, the diagonal and offdiagonal entries of h_{ CNOT } represent the parameters E_{ m }and J_{m,l}, respectively, for the sixlattice sites, and sites 1 and 6 are the auxiliary sites. The gate operation and expected output probabilities are depicted in Fig. 5. It is interesting to note that while the peak of the curve in Fig. 4 is found at 1/9, the fidelity starts to fall significantly only at around 2/9, suggesting the possibility of systems that trade singlegate fidelity for higher success probabilities. The gate operation at the highest fidelity is presented in Fig. 5, depicting the real and imaginary matrix entries in the logical basis (Fig. 5a, b), as well as the photon density evolution (Fig. 5c–f).
Compiling a threequbit primitive
Implementing a quantum algorithm using the scheme presented in this paper will involve several lattice configurations operating in sequence, as gates are sequentially applied in the algorithm. In principle, because the gate set presented in this work is universal, any multiqubit operation can be broken down into a sequence of singlequbit and twoqubit gates, and thus implemented using the gates already presented. However, compiling common multistep operations into a single primitive based on a single, timeindependent Hamiltonian could reduce the possibility of errors arising from dynamic changes to the lattice. As an example, we constructed a threequbit gate, shown in Fig. 6. This gate is useful, for instance, in the 2bit DeutschJozsa algorithm,^{37} performing the oracle for the function f(x,y) = x ⊕ y. (All other oracles for the 2bit Deutsch–Jozsa algorithm are either a simple variation of this oracle or require only singlequbit gates plus at most one CNOT gate.) Our computational approach allowed us to find a set of lattice parameters that realizes the complete threequbit operation in a single gate. We focus on the interacting boson case; as linear optical gates are postselected, finding a multiparticle gate that is of high fidelity and also satiesfies unknown postselection criteria requires a nontrivial amount of work beyond the scope of this article. Figure 6 presents an implementation of this threequbit operation, at a fidelity of 99.8%, using a single, onedimensional sixsite lattice:
with interaction strength Γ = 108.24π. In this case as well, the fidelity could be improved by allowing larger tunneling rates.
Discussion
We have shown that multiparticle continuoustime QWs in one dimension can implement quantum logic gates. For QWs of interacting bosons—a situation that arises for ultracold atoms in optical potentials—QWs yield universal deterministic quantum logic gates. For noninteracting photonic walks, our approach yields physically simpler probabilistic quantum logic gates than previously found. Our results can be implemented in integrated quantum photonic devices and ultracold atom systems that allow for singleparticle manipulation and detection.
Methods
In this section, we detail the numerical methods used to find the gates presented in this work. We used a freesoftware implementation of a variety of numerical optimization algorithms.^{29} This software provides black box optimizers; this enabled us to, with a single specification of cost function and constraints, compare the success and computational cost of a number of different optimization approaches. We found that a randomly seeded global optimization algorithm^{30,31} combined with a gradientfree local algorithm^{32} gave the best performance, both in terms of number of iterations and computational runtime.
Careful selection of the cost function was crucial to the success of this work, and interacted with the choice of optimization algorithms, particularly the local optimizer. Throughout the paper, we define the fidelity of the gate in terms of the Hilbert–Schmidt innerproduct between the target unitary gate operation U_{0} and the unitary operation U generated by the Hamiltonian at a given step of the optimization (restricted to the logical subspace). Specifically, the fidelity is defined to be
with
$$\left\langle {U_0,U} \right\rangle _{\mathrm{C}} = \frac{{{\mathrm{Tr}}(U_0^\dagger U)}}{N},$$where N is the dimension of the logical space (4 for twoqubit gates). This fidelity can be interpreted as a lowerbound average fidelity of the gate.
To be specific about the iterative numerical process, at each step we generated U from a vector corresponding to lattice parameters and calculated F(U_{0}, U). Numerically, we found that minimizing the function 1 − F^{2}, rather than 1 − F, gave superior performance. In the case of the algorithm given in ref. ^{32} the reason for this is clear: the algorithm assumes a quadratic cost function. However, we found that even with algorithms designed for linear cost functions (e.g., ^{38}), convergence was much slower than for the quadratic cost function.
Finally, in order to ensure that U has the same global phase as U_{0} (this is for esthetic purposes, as F(U_{0}, U) is invariant under multiplication by a global phase), we placed a cost on the phase of the matrix element u_{1,1}. We found this to be most efficiently implemented by adding the term sin(arg(u_{1,1}))^{2} to the cost function. This function is quadratic when perturbed about zero, is nonnegative, and is symmetric about nπ for all \(n \in {\Bbb Z}\), making it an ideal candidate function. We verified that the introduction of this additional cost both yielded a U with appropriate phase (see Fig. 2) and did not result in a decreased fidelity compared to optimization without this constraint.
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Source code and data are available from the authors upon reasonable request.
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Acknowledgements
We acknowledge helpful discussions with Terry Orlando, William Oliver, Philipp Preiss and Markus Greiner’s group. Y.L. acknowledges support from the Pappalardo program in Physics. G.R.S. was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. D.E. acknowledges support from AFOSR MURI program under grant number FA95501410052.
Author information
Author notes
Yoav Lahini and Gregory R. Steinbrecher contributed equally to this work.
Affiliations
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
 Yoav Lahini
 & Adam D. Bookatz
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel
 Yoav Lahini
Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA
 Gregory R. Steinbrecher
 & Dirk Englund
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Contributions
Y.L. initiated the project, G.R.S. developed the simulation and optimization code, Y.L. and G.R.S ran the code and analyzed the results, A.D.K. performed theoretical analysis and D.E. supervised the work. All authors participated in writing the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Dirk Englund.
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