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Register a new userQuantum algorithms for spin models and simulable gate sets for quantum computation
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Added by
Maarten Van den Nest
Publication date
2008
fields
Physics
and research's language is
English
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We present elementary mappings between classical lattice models and quantum circuits. These mappings provide a general framework to obtain efficiently simulable quantum gate sets from exactly solvable classical models. For example, we recover and generalize the simulability of Valiants match-gates by invoking the solvability of the free-fermion eight-vertex model. Our mappings furthermore provide a systematic formalism to obtain simple quantum algorithms to approximate partition functions of lattice models in certain complex-parameter regimes. For example, we present an efficient quantum algorithm for the six-vertex model as well as a 2D Ising-type model. We finally show that simulating our quantum algorithms on a classical computer is as hard as simulating universal quantum computation (i.e. BQP-complete).
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Variational quantum algorithms are believed to be promising for solving computationally hard problems and are often comprised of repeated layers of quantum gates. An example thereof is the quantum approximate optimization algorithm (QAOA), an approach to solve combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) systems. Gaining computational power from QAOA critically relies on the mitigation of errors during the execution of the algorithm, which for coherence-limited operations is achievable by reducing the gate count. Here, we demonstrate an improvement of up to a factor of 3 in algorithmic performance as measured by the success probability, by implementing a continuous hardware-efficient gate set using superconducting quantum circuits. This gate set allows us to perform the phase separation step in QAOA with a single physical gate for each pair of qubits instead of decomposing it into two C$Z$-gates and single-qubit gates. With this reduced number of physical gates, which scales with the number of layers employed in the algorithm, we experimentally investigate the circuit-depth-dependent performance of QAOA applied to exact-cover problem instances mapped onto three and seven qubits, using up to a total of 399 operations and up to 9 layers. Our results demonstrate that the use of continuous gate sets may be a key component in extending the impact of near-term quantum computers.
We give efficient quantum algorithms to estimate the partition function of (i) the six vertex model on a two-dimensional (2D) square lattice, (ii) the Ising model with magnetic fields on a planar graph, (iii) the Potts model on a quasi 2D square lattice, and (iv) the Z_2 lattice gauge theory on a three-dimensional square lattice. Moreover, we prove that these problems are BQP-complete, that is, that estimating these partition functions is as hard as simulating arbitrary quantum computation. The results are proven for a complex parameter regime of the models. The proofs are based on a mapping relating partition functions to quantum circuits introduced in [Van den Nest et al., Phys. Rev. A 80, 052334 (2009)] and extended here.
Quantum computers promise dramatic speed ups for many computational tasks. For large-scale quantum computation however, the inevitable coupling of physical qubits to the noisy environment imposes a major challenge for a real-life implementation. A scheme introduced by Gottesmann and Chuang can help to overcome this difficulty by performing universal quantum gates in a fault-tolerant manner. Here, we report a non-trivial demonstration of this architecture by performing a teleportation-based two-qubit controlled-NOT gate through linear optics with a high-fidelity six-photon interferometer. The obtained results clearly prove the involved working principles and the entangling capability of the gate. Our experiment represents an important step towards the feasibility of realistic quantum computers and could trigger many further applications in linear optics quantum information processing.
Presently, one of the most ambitious technological goals is the development of devices working under the laws of quantum mechanics. One prominent target is the quantum computer, which would allow the processing of information at quantum level for purposes not achievable with even the most powerful computer resources. The large-scale implementation of quantum information would be a game changer for current technology, because it would allow unprecedented parallelised computation and secure encryption based on the principles of quantum superposition and entanglement. Currently, there are several physical platforms racing to achieve the level of performance required for the quantum hardware to step into the realm of practical quantum information applications. Several materials have been proposed to fulfil this task, ranging from quantum dots, Bose-Einstein condensates, spin impurities, superconducting circuits, molecules, amongst others. Magnetic molecules are among the list of promising building blocks, due to (i) their intrinsic monodispersity, (ii) discrete energy levels (iii) the possibility of chemical quantum state engineering, and (iv) their multilevel characteristics, leading to the so called Qudits (d > 2), amongst others. Herein we review how a molecular multilevel nuclear spin qubit (or qudit, where d = 4), known as TbPc2, gathers all the necessary requirements to perform as a molecular hardware platform with a first generation of molecular devices enabling even quantum algorithm operations.
To assess whether a gate-based quantum algorithm can be executed successfully on a noisy intermediate-scale quantum (NISQ) device, both complexity and actual value of quantum resources should be considered carefully. Based on quantum phase estimation, we implemente arbitrary controlled rotation of quantum algorithms with a proposed modular method. The proposed method is not limited to be used as a submodule of the HHL algorithm and can be applied to more general quantum machine learning algorithms. Compared with the polynomial-fitting function method, our method only requires the least ancillas and the least quantum gates to maintain the high fidelity of quantum algorithms. The method theoretically will not influence the acceleration of original algorithms. Numerical simulations illustrate the effectiveness of the proposed method. Furthermore, if the corresponding diagonal unitary matrix can be effectively decomposed, the method is also polynomial in time cost.
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