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The LHZ parity embedding is one of the front-running methods for implementing difficult-to-engineer long-range interactions in quantum optimisation problems. Continuous-time quantum walks are a leading approach for solving quantum optimisation problems. Due to them populating excited states, quantum walks can avoid the exponential gap closing problems seen in other continuous-time techniques such as quantum annealing and adiabatic quantum computation (AQC). An important question therefore, is how continuous-time quantum walks perform in combination with the LHZ parity embedding. By numerically simulating continuous-time quantum walks on 4, 5 and 6 logical qubit Sherrington-Kirkpatrick (SK) Ising spin glass instances embedded onto the LHZ parity architecture, we are able to verify the continued efficacy of heuristics used to estimate the optimal hopping rate and the numerical agreement with the theory behind the location of the lower bound of the LHZ parity constraint strength. In addition, by comparing several different LHZ-based decoding methods, we were able to identify post-readout error correction techniques which were able to improve the success probability of the quantum walk., Comment: 16 pages, 12 figures

We present a general method for the implementation of quantum algorithms that optimizes both gate count and circuit depth. Our approach introduces connectivity-adapted CNOT-based building blocks called Parity Twine chains. It outperforms all known state-of-the art methods for implementing prominent quantum algorithms such as the quantum Fourier transform or the Quantum Approximate Optimization Algorithm across a wide range of quantum hardware, including linear, square-grid, hexagonal, ladder and all-to-all connected devices. For specific cases, we rigorously prove the optimality of our approach.

Using the Parity Flow formalism, we show that physical SWAP gates can be eliminated in linear hardware architectures, without increasing the total number of two-qubit operations. This has a significant impact on the execution time of quantum circuits in linear Quantum Charge-Coupled Devices (QCCDs), where SWAP gates are implemented by physically changing the position of the ions. Because SWAP gates are one of the most time-consuming operations in QCCDs, our scheme considerably reduces the runtime of the quantum Fourier transform and the quantum approximate optimization algorithm on all-to-all spin models, compared to circuits generated with standard compilers (TKET and Qiskit). While increasing the problem size (and therefore the number of qubits) typically demands longer runtimes, which are constrained by coherence time, our runtime reduction enables a significant increase in the number of qubits at a given coherence time., Comment: 9 pages, 4 figures. Typos corrected

We propose a scalable encoding of combinatorial optimization problems with arbitrary connectivity, including higher-order terms, on arrays of trapped neutral atoms requiring only a global laser drive. Our approach relies on modular arrangements of a small number of problem-independent gadgets. These gadgets represent maximum-weight independent set (MWIS) problems on unit-disk graphs, which are native to such devices. Instead of programming MWIS weights with site-dependent laser detunings, the scheme relies on systematic placements of auxiliary atoms. We show, that these auxiliary atoms can be simultaneously used for both problem-specific programming and the mitigation of unwanted effects originating from the tails of long-range interactions., Comment: 11 pages, 5 figures

The practical implementation of quantum optimization algorithms on noisy intermediate-scale quantum devices requires accounting for their limited connectivity. As such, the Parity architecture was introduced to overcome this limitation by encoding binary optimization problems onto planar quantum chips. We investigate the performance of the Quantum Approximate Optimization Algorithm on the Parity architecture (Parity QAOA) for solving instances of the signed Max-Cut problem on complete and regular graphs. By comparing the algorithms at fixed circuit depth, we demonstrate that Parity QAOA outperforms conventional QAOA implementations based on SWAP networks. Our analysis utilizes Clifford circuits to estimate lower performance bounds for Parity QAOA for problem sizes that would be otherwise inaccessible on classical computers. For single layer circuits we additionally benchmark the recursive variant of the two algorithms, showing that their performance is equal.

We present a formalism based on tracking the flow of parity quantum information to implement algorithms on devices with limited connectivity without qubit overhead, SWAP operations or shuttling. Instead, we leverage the fact that entangling gates not only manipulate quantum states but can also be exploited to transport quantum information. We demonstrate the effectiveness of this method by applying it to the quantum Fourier transform (QFT) and the Quantum Approximate Optimization Algorithm (QAOA) with $n$ qubits. This improves upon all state-of-the-art implementations of the QFT on a linear nearest-neighbor architecture, resulting in a total circuit depth of ${5n-3}$ and requiring ${n^2-1}$ CNOT gates. For the QAOA, our method outperforms SWAP networks, which are currently the most efficient implementation of the QAOA on a linear architecture. We further demonstrate the potential to balance qubit count against circuit depth by implementing the QAOA on twice the number of qubits using bi-linear connectivity, which approximately halves the circuit depth., Comment: 10 pages, 5 figures

One of the main bottlenecks in solving combinatorial optimization problems with quantum annealers is the qubit connectivity in the hardware. A possible solution for larger connectivty is minor embedding. This techniques makes the geometrical properties of the combinatorial optimization problem, encoded as a Hamiltonian, match the properties of the quantum annealing hardware. The embedding itself is a hard computational problem and therefore heuristic algorithms are required. In this work, we present fixed, modular and scalable embeddings that can be used to embed any combinatorial optimization problem described as an Ising Hamiltonian. These embeddings are the result of an extension of the well-known parity mapping, which has been used in the past to map higher-order Ising Hamiltonians to quadratic Hamiltonians, which are suitable for existing quantum hardware. We show how our new embeddings can be mapped to existing quantum annealers and that the embedded Hamiltonian physical properties match the original Hamiltonian properties.

We present a fault-tolerant universal quantum computing architecture based on a code concatenation of noise-biased qubits and the parity architecture. The parity architecture can be understood as a LDPC code tailored specifically to obtain any desired logical connectivity from nearest neighbor physical interactions. The code layout can be dynamically adjusted to algorithmic requirements on-the-fly. This allows reaching arbitrary code distances and thereby exponential suppression of errors with a universal set of fault-tolerant gates. In addition to the previously explored tool-sets for concatenated cat codes, our approach features parallelizable interactions between arbitrary sets of qubits by directly addressing the parity qubits in the code. The proposed scheme enables codes with less physical qubit overhead compared to the repetition code with the same code distances, while requiring only weight-3 and weight-4 stabilizers and nearest neighbor 2D square-lattice connectivity., Comment: 8 pages

Motivated by the prospect of a two-dimensional square-lattice geometry for semiconductor spin qubits, we explore the realization of the Parity Architecture with quantum dots (QDs). We present sequences of spin shuttling and quantum gates that implement the Parity Quantum Approximate Optimization Algorithm (QAOA) on a lattice constructed of identical unit cells, such that the circuit depth is always constant. We further develop a detailed error model for a hardware-specific analysis of the Parity Architecture and we estimate the errors during one round of Parity QAOA. The model includes a general description of the shuttling errors as a function of the probability distribution function of the valley splitting, which is the main limitation for the performance. We compare our approach to a superconducting transmon qubit chip and we find that with high-fidelity spin shuttling the performance of the spin qubits is competitive or even exceeds the results of the transmons. Finally, we discuss the possibility of decoding the logical quantum state and of quantum error mitigation. We find that already with near-term spin qubit devices a sufficiently low physical error probability can be expected to reliably perform Parity QAOA with a short depth in a regime where the success probability compares favorably to standard QAOA., Comment: Supplemental Material attached, to view please download "Other formats / anc"

In this article, we establish a mathematical framework that utilizes concepts from graph theory to formalize the parity transformation, an encoding strategy for compiling optimization problems on quantum devices. We introduce the transformation as a mapping that encompasses all possible compiled hypergraphs and investigate its uniqueness properties in more detail. Specifically, by introducing so-called loop labelings, we derive an alternative expression of the preimage of any set of compiled hypergraphs under this encoding procedure when all equivalence classes of graphs are being considered. We then deduce equivalent conditions for the injectivity of the parity transformation on any subset of all equivalences classes of graphs. Through concrete examples, we demonstrate that the parity transformation is not an injective mapping, and also introduce an important class of physical layouts and their corresponding set of constraints whose preimage is uniquely determined. In addition, we provide an algorithm which is based on classical algorithms from theoretical computer science and computes a compiled physical layout in this class in polynomial time.

Parity compilation is the challenge of laying out the required constraints for the parity mapping in a local way. We present the first constructive compilation algorithm for the parity architecture using plaquettes for arbitrary higher-order optimization problems. This enables adiabatic protocols, where the plaquette layout can natively be implemented, as well as fully parallelized digital circuits. The algorithm builds a rectangular layout of plaquettes, where in each layer of the rectangle at least one constraint is added. The core idea is that each constraint, consisting of any qubits on the boundary of the rectangle and some new qubits, can be decomposed into plaquettes with a deterministic procedure using ancillas. We show how to pick a valid set of constraints and how this decomposition works. We further give ways to optimize the ancilla count and show how to implement optimization problems with additional constraints., Comment: 8 pages, 5 figures

We present tools and methods to generalize parity compilation to digital quantum computing devices with arbitrary connectivity graphs and construct circuit implementations for the constraint Hamiltonian of higher-order constrained binary optimization problems. In particular, we show how even non-local constraints can be efficiently implemented without expensive SWAP gates. We show how the presented tools can be used to optimize the total circuit depth and CNOT count of the quantum approximate optimization algorithm in the parity architecture and highlight the advantages of the flexible compilation using various examples. We derive the relation between the developed gate sequences and the traditional approach that uses SWAP gates. The result can be applied to improve the implementation of many other non-local operators., Comment: 10 pages, 7 figures

An important aspect that strongly impacts the experimental feasibility of quantum circuits is the ratio of gate times and typical error time scales. Algorithms with circuit depths that significantly exceed the error time scales will result in faulty quantum states and error correction is inevitable. We present a comparison of the theoretical minimal gate time, i.e., the quantum speed limit (QSL), for realistic two- and multi-qubit gate implementations in neutral atoms and superconducting qubits. Subsequent to finding the QSLs for individual gates by means of optimal control theory we use them to quantify the circuit QSL of the quantum Fourier transform and the quantum approximate optimization algorithm. In particular, we analyze these quantum algorithms in terms of circuit run times and gate counts both in the standard gate model and the parity mapping. We find that neutral atom and superconducting qubit platforms show comparable weighted circuit QSLs with respect to the system size., Comment: 20 pages, 5 figures

We present a protocol to encode and decode arbitrary quantum states in the parity architecture with constant circuit depth using measurements, local nearest-neighbor and single-qubit operations only. While this procedure typically requires a quadratic overhead of simultaneous qubit measurements, it allows for a simple and low-depth implementation of logical multi-qubit gates in the parity encoding via code deformation. We discuss how such encoding and decoding schemes can be used to flexibly change the size and shape of the underlying code to enable a more efficient implementation of quantum gates or algorithms. We apply the new findings to the QAOA which leads to a constant depth implementation using local gates at the same optimization performance as the standard, potentially non-local, QAOA approach without the parity encoding. Furthermore, we show that our method can reduce the depth of implementing the quantum Fourier transform by a factor of two when allowing measurements., Comment: 11 pages, 8 figures

We present an encoding and hardware-independent formulation of optimization problems for quantum computing. Using this generalized approach, we present an extensive library of optimization problems and their various derived spin encodings. Common building blocks that serve as a construction kit for building these spin Hamiltonians are identified. This paves the way towards a fully automatic construction of Hamiltonians for arbitrary discrete optimization problems. The presented freedom in the problem formulation is a key step for tailoring optimal spin Hamiltonians for different hardware platforms., Comment: 29 pages, 5 figures

Solving optimization problems on near term quantum devices requires developing error mitigation techniques to cope with hardware decoherence and dephasing processes. We propose a mitigation technique based on the LHZ architecture. This architecture uses a redundant encoding of logical variables to solve optimization problems on fully programmable planar quantum chips. We discuss how this redundancy can be exploited to mitigate errors in quantum optimization algorithms. In the specific context of the quantum approximate optimization algorithm (QAOA), we show that errors can be significantly mitigated by appropriately modifying the objective cost function.

We present a construction for circuits with low gate count and depth, implementing three- and four-body Pauli-Z product operators as they appear in the form of plaquette-shaped constraints in QAOA when using the parity mapping. The circuits can be implemented on any quantum device with nearest-neighbor connectivity on a square-lattice, using only one gate type and one orientation of two-qubit gates at a time. We find an upper bound for the circuit depth which is independent of the system size. The procedure is readily adjustable to hardware-specific restrictions, such as a minimum required spatial distance between simultaneously executed gates, or gates only being simultaneously executable within a subset of all the qubits, for example a single line., Comment: 11 pages, 8 figures

We present a scalable architecture for solving higher-order constrained binary optimization problems on current neutral-atom hardware operating in the Rydberg blockade regime. In particular, we formulate the recently developed parity encoding of arbitrary connected higher-order optimization problems as a maximum-weight independent set (\textsf{MWIS}) problem on disk graphs, that are directly encodable on such devices. Our architecture builds from small \textsf{MWIS} modules in a problem-independent way, crucial for practical scalability., Comment: 10 pages, 7 figures

Approximate counterdiabatic (CD) protocols are a powerful tool to enhance quantum adiabatic processes that allow to reliably manipulate quantum systems on short time scales. However, implementing CD protocols entails the introduction of additional control fields in the Hamiltonian, often associated with highly non-local multi-body interactions. Here, we introduce a novel variational rotated ansatz (RA) to systematically generate experimentally accessible approximate CD protocols. We numerically benchmark our approach on state preparation and adiabatic quantum computing algorithms, and find that using RA protocols significantly enhances their performances.

We demonstrate the applicability of a universal gate set in the parity encoding, which is a dual to the standard gate model, by exploring several quantum gate algorithms such as the quantum Fourier transform and quantum addition. Embedding these algorithms in the parity encoding reduces the circuit depth compared to conventional gate-based implementations while keeping the multiqubit gate counts comparable. We further propose simple implementations of multiqubit gates in tailored encodings and an efficient strategy to prepare graph states., Comment: 9 pages, 6 figures

We propose a universal gate set for quantum computing with all-to-all connectivity and intrinsic robustness to bit-flip errors based on parity encoding. We show that logical controlled phase gate and $R_z$ rotations can be implemented in parity encoding with single-qubit operations. Together with logical $R_x$ rotations, implemented via nearest-neighbor controlled-NOT gates and an $R_x$ rotation, these form a universal gate set. As the controlled phase gate requires only single-qubit rotations, the proposed scheme has advantages for several cornerstone quantum algorithms, e.g., the quantum Fourier transform. We present a method to switch between different encoding variants via partial on-the-fly encoding and decoding., Comment: 9 pages, 6 figures

The parity transformation encodes spin models in the low-energy subspace of a larger Hilbert-space with constraints on a planar lattice. Applying the Quantum Approximate Optimization Algorithm (QAOA), the constraints can either be enforced explicitly, by energy penalties, or implicitly, by restricting the dynamics to the low-energy subspace via the driver Hamiltonian. While the explicit approach allows for parallelization with a system-size-independent circuit depth, the implicit approach shows better QAOA performance. Here we combine the two approaches in order to improve the QAOA performance while keeping the circuit parallelizable. In particular, we introduce a modular parallelization method that partitions the circuit into clusters of subcircuits with fixed maximal circuit depth, relevant for scaling up to large system sizes., Comment: 11 pages, 9 figures

This article proposes quantum processing in an optical lattice, using Rydberg electron's Fermi scattering from ground-state atoms in spin-dependent lattices as a source of interaction. Instead of relying on Rydberg pair potentials, the interaction is controlled by engineering the electron cloud of a sole Rydberg atom. Here we specifically propose the implementation of two prominent multi-qubit gates i.e. the stabilizer-phase operator and the Toffoli gate. The new scheme addresses the main bottleneck in Rydberg quantum simulation by suppressing the population of short-lived Rydberg states over multi-qubit operations. This scheme mitigates different competing infidelity criteria, eliminates unwanted cross-talks, and allows operations in dense atomic lattices. The restoring forces in the molecule type Ryd-Fermi potential preserve the trapping over a long interaction period. The features in the new scheme are of special interest for the implementation of quantum optimization and error correction algorithms.

The preparation of ground states of spin systems is a fundamental operation in quantum computing and serves as the basis of adiabatic quantum computing. This form of quantum computation is subject to the adiabatic theorem which in turn poses a fundamental speed limit. We show that by employing diabatic transitions via counter diabatic driving a less strict requirement on adiabaticity applies. We demonstrate a scaling advantage from local and multi-spin counter diabatic driving in the ground-state fidelity compared to their adiabatic counterpart, for different Ising spin models., Comment: 8 pages, 7 figures

We introduce CircuitQ, an open-source toolbox for the analysis of superconducting circuits implemented in Python. It features the automated construction of a symbolic Hamiltonian of the input circuit and a dynamic numerical representation of the Hamiltonian with a variable basis choice. Additional features include the estimation of the T1 lifetimes of the circuit states under various noise mechanisms. We review previously established circuit quantization methods and formulate them in a way that facilitates the software implementation. The toolbox is then showcased by applying it to practically relevant qubit circuits and comparing it to specialized circuit solvers. Our circuit quantization is applicable to circuit inputs from a large design space, and the software is open-sourced. We thereby add an important resource for the design of new quantum circuits for quantum information processing applications.

There is a large ongoing research effort towards obtaining a quantum advantage in the solution of combinatorial optimization problems on near-term quantum devices. A particularly promising platform for testing and developing quantum optimization algorithms are arrays of trapped neutral atoms, laser-coupled to highly excited Rydberg states. However, encoding combinatorial optimization problems in atomic arrays is challenging due to the limited inter-qubit connectivity given by their native finite-range interactions. Here we propose and analyze a fast, high fidelity four-body Rydberg parity gate, enabling a direct and straightforward implementation of the Lechner-Hauke-Zoller (LHZ) scheme and its recent generalization, the parity architecture, a scalable architecture for encoding arbitrarily connected interaction graphs. Our gate relies on onetime-optimized adiabatic laser pulses and is fully programmable by adjusting two hold-times during operation. We numerically demonstrate an implementation of the quantum approximate optimization algorithm (QAOA) for a small scale test problem. Our approach allows for efficient execution of variational optimization steps with a constant number of system manipulations, independent of the system size, thus paving the way for experimental investigations of QAOA beyond the reach of numerical simulations., Comment: 11 pages

We present benchmarks of the parity transformation for the Quantum Approximate Optimization Algorithm (QAOA). We analyse the gate resources required to implement a single QAOA cycle for real-world scenarios. In particular, we consider random spin models with higher order terms, as well as the problems of predicting financial crashes and finding the ground states of electronic structure Hamiltonians. For the spin models studied our findings imply a significant advantage of the parity mapping compared to the standard gate model. In combination with full parallelizability of gates this has the potential to boost the race for demonstrating quantum advantage.

We introduce parity quantum optimization with the aim of solving optimization problems consisting of arbitrary $k$-body interactions and side conditions using planar quantum chip architectures. The method introduces a decomposition of the problem graph with arbitrary $k$-body terms using generalized closed cycles of a hypergraph. Side conditions of the optimization problem in form of hard constraints can be included as open cycles containing the terms involved in the side conditions. The generalized parity mapping thus circumvents the need to translate optimization problems to a quadratic unconstrained binary optimization problem (QUBO) and allows for the direct encoding of higher-order constrained binary optimization problems (HCBO) on a square lattice and full parallelizability of gates.

Constraints make hard optimization problems even harder to solve on quantum devices because they are implemented with large energy penalties and additional qubit overhead. The parity mapping, which has been introduced as an alternative to the spin encoding, translates the problem to a representation using only parity variables that encodes products of spin variables. In combining exchange interaction and single spin flip terms in the parity representation, constraints on sums and products of arbitrary k-body terms can be implemented without additional overhead in two-dimensional quantum systems.

In order to treat all-to-all connected quadratic binary optimization problems (QUBO) with hardware quantum annealers, an embedding of the original problem is required due to the sparsity of the hardware's topology. Embedding fully-connected graphs - typically found in industrial applications - incurs a quadratic space overhead and thus a significant overhead in the time to solution. Here we investigate this embedding penalty of established planar embedding schemes such as minor embedding on a square lattice, minor embedding on a Chimera graph, and the Lechner-Hauke-Zoller scheme using simulated quantum annealing on classical hardware. Large-scale quantum Monte Carlo simulation suggest a polynomial time-to-solution overhead. Our results demonstrate that standard analog quantum annealing hardware is at a disadvantage in comparison to classical digital annealers, as well as gate-model quantum annealers and could also serve as benchmark for improvements of the standard quantum annealing protocol., Comment: 10 pages, 10 figures, 1 table

Resonant excitation of the biexciton state in an InAsP quantum dot by a phase-coherent pair of picosecond pulses allows preparing time-bin entangled pairs of photons via the biexciton-exciton cascade. We show that this scheme can be implemented for a dot embedded in an InP nanowire. The underlying physical mechanisms can be represented and quantitatively analyzed by an effective three-level open system master equation. Simulation parameters including decay and intensity dependent dephasing rates are extracted from experimental data, which in turn let us predict the resulting entanglement and optimal operating conditions.

In order to qualify quantum algorithms for industrial NP-Hard problems, comparing them to available polynomial approximate classical algorithms and not only to exact ones -- exponential by nature -- , is necessary. This is a great challenge as, in many cases, bounds on the reachable approximation ratios exist according to some highly-trusted conjectures of Complexity Theory. An interesting setup for such qualification is thus to focus on particular instances of these problems known to be "less difficult" than the worst-case ones and for which the above bounds can be outperformed: quantum algorithms should perform at least as well as the conventional approximate ones on these instances, up to very large sizes. We present a case study of such a protocol for two industrial problems drawn from the strongly developing field of smart-charging of electric vehicles. Tailored implementations of the Quantum Approximate Optimization Algorithm (QAOA) have been developed for both problems, and tested numerically with classical resources either by emulation of Pasqal's Rydberg atom based quantum device or using Atos Quantum Learning Machine. In both cases, quantum algorithms exhibit the same approximation ratios than conventional approximation algorithms, or improve them. These are very encouraging results, although still for instances of limited size as allowed by studies on classical computing resources. The next step will be to confirm them on larger instances, on actual devices, and for more complex versions of the problems addressed.

We introduce a two-parameter approximate counter-diabatic term into the Hamiltonian of the transverse-field Ising model for quantum annealing to accelerate convergence to the solution, generalizing an existing single-parameter approach. The protocol is equivalent to unconventional diabatic control of the longitudinal and transverse fields in the transverse-field Ising model and thus makes it more feasible for experimental realization than an introduction of new terms such as non-stoquastic catalysts toward the same goal of performance enhancement. We test the idea for the $p$-spin model with $p=3$, which has a first-order quantum phase transition, and show that our two-parameter approach leads to significantly larger ground-state fidelity and lower residual energy than those by traditional quantum annealing as well as by the single-parameter method. We also find a scaling advantage in terms of the time to solution as a function of the system size in a certain range of parameters as compared to the traditional methods., Comment: 14 pages, 6 figures

As a means to solve optimization problems using quantum computers, the problem is typically recast into a Ising spin model whose ground-state is the solution of the optimization problem. An alternative to the Ising formulation is the Lechner-Hauke-Zoller model, which has the form of a lattice gauge model with nearest neighbor 4-body constraints. Here we introduce a method to find the minimal strength of the constraints which are required to conserve the correct ground-state. Based on this, we derive upper and lower bounds for the minimal constraints strengths. We find that depending on the problem class, the exponent ranges from linear $\alpha \propto 1$ to quadratic $\alpha \propto 2$ scaling with the number of logical qubits.

Quantum refrigerators pump heat from a cold to a hot reservoir. In the few-particle regime, counter-diabatic (CD) driving of, originally adiabatic, work-exchange strokes is a promising candidate to overcome the bottleneck of vanishing cooling power. Here, we present a finite-time many-body quantum refrigerator that yields finite cooling power at high coefficient of performance, that considerably outperforms its non-adiabatic counterpart. We employ multi-spin CD driving and numerically investigate the scaling behavior of the refrigeration performance with system size. We further prove that optimal refrigeration via the exact CD protocol is a catalytic process., Comment: 13 pages, 5 figures

Quantum heat engines are modeled by thermodynamic cycles with quantum-mechanical working media. Since high engine efficiencies require adiabaticity, a major challenge is to yield a nonvanishing power output at finite cycle times. Shortcuts to adiabaticity using counter-diabatic (CD) driving may serve as a means to speed up such, otherwise infinitely long, cycles. We introduce local approximate CD protocols for many-body spin quantum heat engines and show that this method improves the efficiency and power for finite cycle times considerably. The protocol does not require a priori knowledge of the system eigenstates and is thus realistic in experiments., Comment: 15 pages, 7 figures

Clustering algorithms are a cornerstone of machine learning applications. Recently, a quantum algorithm for clustering based on the k-means algorithm has been proposed by Kerenidis, Landman, Luongo and Prakash. Based on their work, we propose a quantum expectation-maximization (EM) algorithm for Gaussian mixture models (GMMs). The robustness and quantum speedup of the algorithm is demonstrated. We also show numerically the advantage of GMM over k-means for non-trivial cluster data., Comment: 10 pages, 9 figures

We study the zero-temperature phase diagram of the Lechner-Hauke-Zoller model. An analytic expression for the free-energy and critical coefficients for finite-size systems and in the thermodynamic limit are derived and numerically verified. With the aim to improve standard quantum annealing, we introduce an inhomogeneously driven transverse field with an additional time-dependent parameter that allows one to evade the first-order quantum phase transition and, thus, improve the efficiency of the ground-state preparation considerably.

Quantum annealing is a computing paradigm that has the ambitious goal of efficiently solving large-scale combinatorial optimization problems of practical importance. However, many challenges have yet to be overcome before this goal can be reached. This perspectives article first gives a brief introduction to the concept of quantum annealing, and then highlights new pathways that may clear the way towards feasible and large scale quantum annealing. Moreover, since this field of research is to a strong degree driven by a synergy between experiment and theory, we discuss both in this work. An important focus in this article is on future perspectives, which complements other review articles, and which we hope will motivate further research., Comment: 37 pages, 9 figures

We present a protocol to store a polynomial number of arbitrary bit strings, encoded as spin configurations, in the approximately degenerate low-energy manifold of an all-to-all connected Ising spin glass. The iterative protocol is inspired by machine learning techniques utilizing $k$-local Hopfield networks trained with $k$-local Hebbian learning and unlearning. The trained Hamiltonian is the basis of a quantum state-preparation scheme to create quantum many-body superpositions with tunable squared amplitudes using resources available in near term experiments. We find that the number of configurations that can be stored in the ground states and thus turned into superposition scales with the $k$-locality of the Ising interaction., Comment: published version, 11 pages, 6 figures

We present a coherent counter-diabatic quantum protocol to prepare ground states in the lattice gauge mapping of all-to-all Ising models (LHZ) with considerably enhanced final ground state fidelity compared to a quantum annealing protocol. We make use of a variational method to find approximate counter-diabatic Hamiltonians that has recently been introduced by Sels and Polkovnikov [Proc. Natl. Acad. Sci. 114, 3909 (2017)]. The resulting additional terms in our protocol are time-dependent local on-site y-magnetic fields. These additional Hamiltonian terms do not increase the minimal energy gap, but instead rely on coherent phase maximization. A single free parameter is introduced which is optimized via classical updates. The protocol consists only of local and nearest-neighbor terms which makes it attractive for implementations in near term experiments., Comment: 11 pages, 9 figures

The N-queens problem is to find the position of N queens on an N by N chess board such that no queens attack each other. The excluded diagonals N-queens problem is a variation where queens cannot be placed on some predefined fields along diagonals. This variation is proven NP-complete and the parameter regime to generate hard instances that are intractable with current classical algorithms is known. We propose a special purpose quantum simulator that implements the excluded diagonals N-queens completion problem using atoms in an optical lattice and cavity-mediated long-range interactions. Our implementation has no overhead from the embedding allowing to directly probe for a possible quantum advantage in near term devices for optimization problems., Comment: 22 pages, 8 figures

The quantum approximate optimization algorithm (QAOA) has been introduced as a heuristic digital quantum computing scheme to find approximate solutions of combinatorial problems with shallow circuits. We present a scheme to parallelize this approach for arbitrary all-to-all connected problem graphs in a layout of quantum bits (qubits) with nearest neighbor interactions. The protocol consisting of single qubit operations that encode the optimization problem and all interactions are problem-independent pair-wise CNOT gates among nearest neighbors. This allows for a parallelizable implementation in quantum devices with a square lattice geometry. The basis of this proposal is a lattice gauge model which also introduces additional parameters and protocols for QAOA to improve the efficiency.