We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.
We propose an adiabatic quantum algorithm capable of factorizing numbers, using fewer qubits than Shor's algorithm. We implement the algorithm in an NMR quantum information processor and experimentally factorize the number 21. Numerical simulations indicate that the running time grows only quadratically with the number of qubits.
In this work Controlled phase shift gates are implemented on a qaudrupolar system, by using non-adiabatic geometric phases. A general procedure is given, for implementing controlled phase shift gates in an 'N' level system. The utility of such controlled phase shift gates, is demonstrated here by implementing 3-qubit Deutsch-Jozsa algorithm on a 7/2 quadrupolar nucleus oriented in a liquid crystal matrix.
Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover's search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, using the local adiabatic evolution, the algorithms given by J. Roland and N. J. Cerf for Grover's search [ Phys. Rev. A. {\bf 65} 0423…
Recent work at Fraunhofer FKIE shows that Morefield's method for multiple target data association can in theory be solved on an adiabatic quantum computer. The present paper validates the theory and examines the significant limitations of currently available adiabatic quantum computers for solving the data association problem. The limitations of such architectures are both theoretical and practical in nature, and both are discussed. The data association problem is formulated as a quadratic unconstrained binary optimization (QUBO) problem; consequently, much of the discussion is relevant to oth…
The Toffoli gate is an essential logic element, which permits implementation of a reversible processor. It is of relevance both for classical as well as quantum logics. We propose and theoretically study all-optical implementations of three-bit and four-bit Toffoli gates by application of adiabatic population transfer techniques. For a three-bit Toffoli gate we use variants of stimulated Raman adiabatic passage (STIRAP) processes in a $Λ$-type level scheme, driven by two laser pulses at sufficiently large detunings. For the implementation of a four-bit Toffoli gate, we apply reversible adiabat…
A cavity QED implementation of the non-adiabatic holonomic quantum computation in decoherence-free subspaces is proposed with nitrogen-vacancy centers coupled commonly to the whispering-gallery mode of a microsphere cavity, where a universal set of quantum gates can be realized on the qubits. In our implementation, with the assistant of the appropriate driving fields, the quantum evolution is insensitive to the cavity field state, which is only virtually excited. The implemented non-adiabatic holonomies, utilizing optical transitions in the Λ type of three-level configuration of the nitrogen-v…
Non-adiabatic holonomic quantum computation has received increasing attention due to its robustness against control errors. However, all the previous schemes have to use at least two sequentially implemented gates to realize a general one-qubit gate. In this paper, we put forward a novelty scheme by which one can directly realize an arbitrary holonomic one-qubit gate with a single-shot implementation, avoiding the extra work of combining two gates into one. Based on a three-level model driven by laser pulses, we show that any single-qubit holonomic gate can be realized by varying the detuning,…
We propose a novel non-Hermitian adiabatic quantum optimization algorithm. One of the new ideas is to use a non-Hermitian auxiliary "initial'' Hamiltonian that provides an effective level repulsion for the main Hamiltonian. This effect enables us to develop an adiabatic theory which determines ground state much more efficiently than Hermitian methods.
We examine a proposed auction using quantum states to represent bids and distributed adiabatic search to find the winner. When the auctioneer follows the protocol, the final measurement giving the outcome of the auction also destroys the bid states, thereby preserving privacy of losing bidders. We describe how a dishonest auctioneer could alter the protocol to violate this privacy guarantee, and present methods by which bidders can counter such attacks. We also suggest possible quantum circuit implementations of the auctions protocol, and quantum circuits to perpetrate and to counter attacks b…
It has been recently argued that adiabatic quantum optimization would fail in solving NP-complete problems because of the occurrence of exponentially small gaps due to crossing of local minima of the final Hamiltonian with its global minimum near the end of the adiabatic evolution. Using perturbation expansion, we analytically show that for the NP-hard problem of maximum independent set there always exist adiabatic paths along which no such crossings occur. Therefore, in order to prove that adiabatic quantum optimization fails for any NP-complete problem, one must prove that it is impossible t…
Quantum annealing is guaranteed to find the ground state of optimization problems in the adiabatic limit. Recent work [Phys. Rev. X 6, 031010 (2016)] has found that for some barrier tunneling problems, quantum annealing can be run much faster than is adiabatically required. Specifically, an $n$-qubit optimization problem was presented for which a non-adiabatic, or diabatic, annealing algorithm requires only constant runtime, while an adiabatic annealing algorithm requires a runtime polynomial in $n$. Here we show that this non-adiabatic speed-up is a direct result of a specific symmetry in the…
We explore to what extent path-integral quantum Monte Carlo methods can efficiently simulate the tunneling behavior of quantum adiabatic optimization algorithms. Specifically we look at symmetric cost functions defined over n bits with a single potential barrier that a successful optimization algorithm will have to tunnel through. The height and width of this barrier depend on n, and by tuning these dependencies, we can make the optimization algorithm succeed or fail in polynomial time. In this article we compare the strength of quantum adiabatic tunneling with that of path-integral quantum Mo…
Quantum fluctuations driven by non-stoquastic Hamiltonians have been conjectured to be an important and perhaps essential missing ingredient for achieving a quantum advantage with adiabatic optimization. We introduce a transformation that maps every non-stoquastic adiabatic path ending in a classical Hamiltonian to a corresponding stoquastic adiabatic path by appropriately adjusting the phase of each matrix entry in the computational basis. We compare the spectral gaps of these adiabatic paths and find both theoretically and numerically that the paths based on non-stoquastic Hamiltonians have …
Adiabatic quantum computing is a promising route to the computational power afforded by quantum information processing. The recent availability of adiabatic hardware has raised challenging questions about how to evaluate adiabatic quantum optimization programs. Processor behavior depends on multiple steps to synthesize an adiabatic quantum program, which are each highly tunable. We present an integrated programming and development environment for adiabatic quantum optimization called JADE that provides control over all the steps taken during program synthesis. JADE captures the workflow needed…
Most experimental and theoretical studies of adiabatic optimization use stoquastic Hamiltonians, whose ground states are expressible using only real nonnegative amplitudes. This raises a question as to whether classical Monte Carlo methods can simulate stoquastic adiabatic algorithms with polynomial overhead. Here, we analyze diffusion Monte Carlo algorithms. We argue that, based on differences between L1 and L2 normalized states, these algorithms suffer from certain obstructions preventing them from efficiently simulating stoquastic adiabatic evolution in generality. In practice however, we o…
Quantum adiabatic passages can be greatly accelerated by a suitable control field, called a counter-diabatic field, which varies during the scan through resonance. Here, we implement this technique on the electron spin of a single nitrogen-vacancy center in diamond. We demonstrate two versions of this scheme. The first follows closely the procedure originally proposed by Demirplak and Rice (J. Phys. Chem. A 107, 9937 (2003)). In the second scheme, we use a control field whose amplitude is constant, but its phase varies with time. This version, which we call the rapid-scan approach, allows an e…
It was recently shown that, for solving NP-complete problems, adiabatic paths always exist without finite-order perturbative crossings between local and global minima, which could lead to anticrossings with exponentially small energy gaps if present. However, it was not shown whether such a path could be found easily. Here, we give a simple construction that deterministically eliminates all such anticrossings in polynomial time, space, and energy, for any Ising models with polynomial final gap. Thus, in order for adiabatic quantum optimization to require exponential time to solve any NP-comple…
Non-adiabatic holonomic quantum computation in decoherence-free subspaces protects quantum information from control imprecisions and decoherence. For the non-collective decoherence that each qubit has its own bath, we show the implementations of two non-commutable holonomic single-qubit gates and one holonomic nontrivial two-qubit gate that compose a universal set of non-adiabatic holonomic quantum gates in decoherence-free-subspaces of the decoupling group, with an encoding rate of $\frac{N-2}{N}$. The proposed scheme is robust against control imprecisions and the non-collective decoherence, …
We analyze the complexity of the quantum optimization algorithm based on adiabatic evolution for the set partition problem. We introduce a cost function defined on a logarithmic scale of the partition residues so that the total number of values of the cost function is of the order of the problem size. We simulate the behavior of the algorithm by numerical solution of the time-dependent Schroedinger equation as well as the stationary equation for the adiabatic eigenvalues. The numerical results for the time-dependent quantum evolution indicate that the complexity of the algorithm scales exponen…
Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as…
Most research regarding quantum adiabatic optimization has focused on stoquastic Hamiltonians, whose ground states can be expressed with only real, nonnegative amplitudes. This raises the question of whether classical Monte Carlo algorithms can efficiently simulate quantum adiabatic optimization with stoquastic Hamiltonians. Recent results have given counterexamples in which path integral and diffusion Monte Carlo fail to do so. However, most adiabatic optimization algorithms, such as for solving MAX-k-SAT problems, use k-local Hamiltonians, whereas our previous counterexample for diffusion Mo…
We lay the foundation for a benchmarking methodology for assessing current and future quantum computers. We pose and begin addressing fundamental questions about how to fairly compare computational devices at vastly different stages of technological maturity. We critically evaluate and offer our own contributions to current quantum benchmarking efforts, in particular those involving adiabatic quantum computation and the Adiabatic Quantum Optimizers produced by D-Wave Systems, Inc. We find that the performance of D-Wave's Adiabatic Quantum Optimizers scales roughly on par with classical approac…
The quantum approximate optimization algorithm (QAOA) is a near-term hybrid algorithm intended to solve combinatorial optimization problems, such as MaxCut. QAOA can be made to mimic an adiabatic schedule, and in the $p\to\infty$ limit the final state is an exact maximal eigenstate in accordance with the adiabatic theorem. In this work, the connection between QAOA and adiabaticity is made explicit by inspecting the regime of $p$ large but finite. By connecting QAOA to counterdiabatic (CD) evolution, we construct CD-QAOA angles which mimic a counterdiabatic schedule by matching Trotter "error" …
It is believed that the presence of anticrossings with exponentially small gaps between the lowest two energy levels of the system Hamiltonian, can render adiabatic quantum optimization inefficient. Here, we present a simple adiabatic quantum algorithm designed to eliminate exponentially small gaps caused by anticrossings between eigenstates that correspond with the local and global minima of the problem Hamiltonian. In each iteration of the algorithm, information is gathered about the local minima that are reached after passing the anticrossing non-adiabatically. This information is then used…
We have developed an adiabatic Abelian geometric quantum computation strategy based on the non-degenerate energy eigenstates in (but not limited to) superconducting phase-qubit systems. The fidelity of the designed quantum gate was evaluated in the presence of simulated thermal fluctuation in superconducting phase qubits circuit and was found to be rather robust against the random errors. In addition, it was elucidated that the Berry phase in the designed adiabatic evolution may be detected directly via the quantum state tomography developed for superconducting qubits.
A scalable superconducting architecture for adiabatic quantum computers is proposed. The architecture is based on time-independent, nearest-neighbor interqubit couplings: it can handle any problem in the class NP even in the presence of measurement errors, noise, and decoherence. The implementation of this architecture with superconducting persistent-current qubits and the natural robustness of such an implementation to manufacturing imprecision and decoherence are discussed.
We provide and analyze examples that counter the widely made claim that tunneling is needed for a quantum speedup in optimization problems. The examples belong to the class of perturbed Hamming-weight optimization problems. In one case, featuring a plateau in the cost function in Hamming weight space, we find that the adiabatic dynamics that make tunneling possible, while superior to simulated annealing, result in a slowdown compared to a diabatic cascade of avoided level-crossings. This, in turn, inspires a classical spin vector dynamics algorithm that is at least as efficient for the plateau…
We present the results of a numerical study, with 20 qubits, of the performance of the Quantum Adiabatic Algorithm on randomly generated instances of MAX 2-SAT with a unique assignment that maximizes the number of satisfied clauses. The probability of obtaining this assignment at the end of the quantum evolution measures the success of the algorithm. Here we report three strategies which consistently increase the success probability for the hardest instances in our ensemble: decreasing the overall evolution time, initializing the system in excited states, and adding a random local Hamiltonian …
It is known that high intensity fields are usually required to implement shortcuts to adiabaticity via Transitionless Quantum Driving (TQD). Here, we show that this requirement can be relaxed by exploiting the gauge freedom of generalized TQD, which is expressed in terms of an arbitrary phase when mimicking the adiabatic evolution. We experimentally investigate the performance of generalized TQD in comparison with both traditional TQD and adiabatic dynamics. By using a $^{171}$Yb$^+$ trapped ion hyperfine qubit, we implement a Landau-Zener adiabatic Hamiltonian and its (traditional and general…
Many artificial intelligence (AI) problems naturally map to NP-hard optimization problems. This has the interesting consequence that enabling human-level capability in machines often requires systems that can handle formally intractable problems. This issue can sometimes (but possibly not always) be resolved by building special-purpose heuristic algorithms, tailored to the problem in question. Because of the continued difficulties in automating certain tasks that are natural for humans, there remains a strong motivation for AI researchers to investigate and apply new algorithms and techniques …
In previous implementations of adiabatic quantum algorithms using spin systems, the average Hamiltonian method with Trotter's formula was conventionally adopted to generate an effective instantaneous Hamiltonian that simulates an adiabatic passage. However, this approach had issues with the precision of the effective Hamiltonian and with the adiabaticity of the evolution. In order to address these, we here propose and experimentally demonstrate a novel scheme for adiabatic quantum computation by using the intrinsic Hamiltonian of a realistic spin system to represent the problem Hamiltonian whi…
We propose a non-convex training objective for robust binary classification of data sets in which label noise is present. The design is guided by the intention of solving the resulting problem by adiabatic quantum optimization. Two requirements are imposed by the engineering constraints of existing quantum hardware: training problems are formulated as quadratic unconstrained binary optimization; and model parameters are represented as binary expansions of low bit-depth. In the present work we validate this approach by using a heuristic classical solver as a stand-in for quantum hardware. Testi…
We discuss applications of shortcuts to adiabaticity (STA) to adiabatic quantum computation. After reviewing the fundamental properties and the present status of STA from the author's personal point of view, we apply the method to the adiabatic algorithm of the Grover's problem. We discuss two possible implementations of STA for the adiabatic quantum computations: the method of quantum adiabatic brachistochrone and the Lewis-Riesenfeld invariant-based inverse engineering.
With progress in quantum technology more sophisticated quantum annealing devices are becoming available. While they offer new possibilities for solving optimization problems, their true potential is still an open question. As the optimal design of adiabatic algorithms plays an important role in their assessment, we illustrate the aspects and challenges to consider when implementing optimization problems on quantum annealing hardware based on the example of the traveling salesman problem (TSP). We demonstrate that tunneling between local minima can be exponentially suppressed if the quantum dyn…
Mechanism of Efficacy in QAOA for Random k-SAT: From Adiabatic Manifold to Sublinear Parameter Optimization
School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；Key Laboratory of Computer Network and Information Integration (Southeast University), Ministry of Education（教育部计算机网络与信息集成重点实验室（东南大学））
Adiabatic quantum computing and optimization have garnered much attention recently as possible models for achieving a quantum advantage over classical approaches to optimization and other special purpose computations. Both techniques are probabilistic in nature and the minimum gap between the ground state and first excited state of the system during evolution is a major factor in determining the success probability. In this work we investigate a strategy for increasing the minimum gap and success probability by introducing intermediate Hamiltonians that modify the evolution path between initia…
Implementation of quantum logical gates for multilevel system is demonstrated through decoherence control under the quantum adiabatic method using simple phase modulated laser pulses. We make use of selective population inversion and Hamiltonian evolution with time to achieve such goals robustly instead of the standard unitary transformation language.
The viability of non-stoquastic catalyst Hamiltonians to deliver consistent quantum speedups in quantum adiabatic optimization remains an open question. The infinite-range ferromagnetic $p$-spin model is a rare example exhibiting an exponential advantage for non-stoquastic catalysts over its stoquastic counterpart. We revisit this model and note how the incremental changes in the ground state wavefunction give an indication of how the non-stoquastic catalyst provides an advantage. We then construct two new examples that exhibit an advantage for non-stoquastic catalysts over stoquastic catalyst…
Topological orders are exotic phases of matter existing in strongly correlated quantum systems, which are beyond the usual symmetry description and cannot be distinguished by local order parameters. Here we report an experimental quantum simulation of the Wen-plaquette spin model with different topological orders in a nuclear magnetic resonance system, and observe the adiabatic transition between two $Z_2$ topological orders through a spin-polarized phase by measuring the nonlocal closed-string (Wilson loop) operator. Moreover, we also measure the entanglement properties of the topological ord…
In this paperwe propose two theoretical schemes for implementation of quantum phase gates by engineering the phase-sensitive dark state of two atoms subjected to Rydberg-Rydberg interaction. Combining the conventional adiabatic techniques and currently developed approaches of phase control, a feasible proposal for implementation of a geometric phase gate is presented, where the conditional phase shift (Berry phase) is achieved by adiabatically and cyclically changing the parameters of the driving fields. Here we find that the geometric phase acquired is related to the way how the relative phas…
We investigate the efficiency of Quantum Adiabatic Optimization when overcoming potential barriers to get from a local to a global minimum. Specifically we look at n qubit systems with symmetric cost functions f:{0, 1}^n->R where the ground state must tunnel through a potential barrier of width n^a and height n^b. By the quantum adiabatic theorem the time delay sufficient to ensure tunneling grows quadratically with the inverse spectral gap during this tunneling process. We analyze barrier sizes with 1/2 < a + b and a < 1/2 and show that the minimum gap scales polynomially as n^{1/2-a-b} when …
Recently, geometric phases, which is fault tolerate to certain errors intrinsically due to its geometric property, are getting considerable attention in quantum computing theoretically. So far, only one experiment about adiabatic geometric gate with NMR through Berry phase has been reported. However, there are two drawbacks in it. First, the adiabatic condition of Berry phase makes such gate very slowly. Second, the extra operation to eliminate the dynamic phase. As we know, geometric phase can exist both adiabatic(Berry phase) and nonadiabatic(Aharonov-Anandan phase). In this letter, we repor…
We develop a resource efficient method by which the ground-state of an arbitrary k-local, optimization Hamiltonian can be encoded as the ground-state of a (k-1)-local optimization Hamiltonian. This result is important because adiabatic quantum algorithms are often most easily formulated using many-body interactions but experimentally available interactions are generally 2-body. In this context, the efficiency of a reduction gadget is measured by the number of ancilla qubits required as well as the amount of control precision needed to implement the resulting Hamiltonian. First, we optimize met…
We propose a protocol for quantum adiabatic optimization, whereby an intermediary Hamiltonian that is diagonal in the computational basis is turned on and off during the interpolation. This `diagonal catalyst' serves to bias the energy landscape towards a given spin configuration, and we show how this can remove the first-order phase transition present in the standard protocol for the ferromagnetic $p$-spin and the Weak-Strong Cluster problems. The success of the protocol also makes clear how it can fail: biasing the energy landscape towards a state only helps in finding the ground state if th…
Quantum computing promises significant improvements of computation capabilities in various fields such as machine learning and complex optimization problems. Recent technological advancements suggest that the adiabatic quantum computing ansatz may soon see practical applications. In this work, we adopt this computation paradigm to develop a quantum computation based solver of the well-known weapon target assignment problem, an NP-hard nonlinear integer programming optimization task. The feasibility of the presented model is demonstrated by numerical simulation of the adiabatic evolution of a s…
We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. We show that for problems that have exponentially large number of local minima close to the global minimum, the gap becomes exponentially small making the computation time exponentially long. The quantum advantage of adiabatic quantum computation may then be accessed only via the local adiabatic evolution, which requires phase coherence throughout the evolution and knowledge of the spectrum. Such problems, therefore, are not s…
Quantum computers with Kerr-nonlinear parametric oscillators (KPOs) have recently been proposed by the author and others. Quantum computation using KPOs is based on quantum adiabatic bifurcations of the KPOs, which lead to quantum superpositions of coherent states, such as Schrodinger cat states. Therefore, these quantum computers are referred to as "quantum bifurcation machines (QbMs)." QbMs can be used for qauntum adiabatic optimization and universal quantum computation. Superconducting circuits with Josephson junctions, Josephson parametric oscillators (JPOs) in particular, are promising fo…
We propose a robust scheme involving atoms fixed in an optical cavity to directly implement the universal controlled-unitary gate. The present technique based on adiabatic passage uses novel dark states well suited for the controlled-rotation operation. We show that these dark states allow the robust implementation of a gate that is a generalisation of the controlled-unitary gate to the case where the control qubit can be selected to be an arbitrary state. This gate has potential applications to the rapid implementation of quantum algorithms such as of the projective measurement algorithm. Thi…
We implement a non-adiabatic universal set of holonomic quantum gates based on abelian holonomies using dynamical invariants, by Lie-algebraic methods. Unlike previous implementations, presented scheme does not rely on secondary methods such as double-loop or spin-echo and avoids associated experimental difficulties. It turns out that such gates exist purely in the non-adiabatic regime for these systems.
Hopfield networks are a variant of associative memory that recall information stored in the couplings of an Ising model. Stored memories are fixed points for the network dynamics that correspond to energetic minima of the spin state. We formulate the recall of memories stored in a Hopfield network using energy minimization by adiabatic quantum optimization (AQO). Numerical simulations of the quantum dynamics allow us to quantify the AQO recall accuracy with respect to the number of stored memories and the noise in the input key. We also investigate AQO performance with respect to how memories …
The non-adiabatic geometric quantum computation (NGQC) has attracted a lot of attention for noise-resilient quantum control. However, previous implementations of NGQC require long evolution paths that make them more vulnerable to incoherent errors than their dynamical counterparts.In this work, we experimentally realize a universal short-path non-adiabatic geometric gate set (SPNGQC) with a 2-times shorter evolution path on a superconducting quantum processor. Characterizing with both quantum process tomography and randomized benchmarking methods, we report an average single-qubit gate fidelit…
We extend the family of problems that may be implemented on an adiabatic quantum optimizer (AQO). When a quadratic optimization problem has at least one set of discrete controls and the constraints are linear, we call this a quadratic constrained mixed discrete optimization (QCMDO) problem. QCMDO problems are NP-hard, and no efficient classical algorithm for their solution is known. Included in the class of QCMDO problems are combinatorial optimization problems constrained by a linear partial differential equation (PDE) or system of linear PDEs. An essential complication commonly encountered i…
