We propose a polynomial-time algorithm for simulation of the class of pairing Hamiltonians, e.g., the BCS Hamiltonian, on an NMR quantum computer. The algorithm adiabatically finds the low-lying spectrum in the vicinity of the gap between ground and first excited states, and provides a test of the applicability of the BCS Hamiltonian to mesoscopic superconducting systems, such as ultra-small metallic grains.
Comment on the Letter ``Polynomial-Time Simulation of Pairing Models on a Quantum Computer'', L. A. Wu, M. S. Byrd and D. A. Lidar, Phys. Rev. Lett. 89, 057904 (2002).
We give out a proposal of quantum simulation of pairing model on an NMR quantum computer. In our proposal, we choose an appropriate initial state which can be easily prepared in experiment. Making use of feature of NMR measure and the technology of the second (discrete) Fourier transformation, our theoretical scheme can obtain the spectrum of paring model in principle. We concretely discuss the case in the concerned subspaces of pairing model and then, as an example, give out a simple initial state to get the gap of two the lowest energy levels in the given subspace. The quantum simulation to …
We have developed a concrete quantum simulation scheme and experimentally simulated a pairing model on an NMR quantum computer. The design of our experiment includes choosing an appropriate initial state in order to make our scheme scalable in near future, and the accomplishment of our experiment makes use of twice Fourier transforms so that our method is applicable to other physical models. Our results show that the experimental simulation can give the spectrum of the simulated Hamiltonian. Consequently, the potential power of a quantum computer on the simulation of complex physical systems i…
We discuss possibilities of utilizing superconductors with Cooper condensates in triplet pairing states (where the spin of condensate pairs is S=1) for practical realization of quantum computers. Superconductors with triplet pairing condensates have features that are unique and cannot be found in the usual (singlet pairing, S=0) superconductors. The symmetry of the order parameter in some triplet superconductors (e.g., ruthenates) corresponds to doubly-degenerate chiral states. These states can serve as qubit base states for quantum computing.
We propose a new simulation computational method to solve the reduced BCS Hamiltonian based on spin analogy and submatrix diagonalization. Then we further apply this method to solve superconducting energy gap and the results are well consistent with those obtained by Bogoliubov transformation method. The exponential problem of 2^{N}-dimension matrix is reduced to the polynomial problem of N-dimension matrix. It is essential to validate this method on a real quantum
A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For $k$ pairs of valence nucleons on $n$ different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one (k+1)x(k+1) and one (n-1)x(k+1) matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, th…
We propose a way to solve BCS-type pairing model by to exactly solve its spin-analogy in the subspace. The advantages of our method are to avoid to directly deal with the approximate procedure and to transfer an exponentially complicated problem to a polynomial problem. Moreover, it builds the theoretical foundation of quantum simulation of pairing model and can be used to check the precision of quantum simulation. It is also helpful to understand the many-body quantum theory.
Although a universal quantum computer is still far from reach, the tremendous advances in controllable quantum devices, in particular with solid-state systems, make it possible to physically implement "quantum simulators". Quantum simulators are physical setups able to simulate other quantum systems efficiently that are intractable on classical computers. Based on solid-state qubit systems with various types of nearest-neighbor interactions, we propose a complete set of algorithms for simulating pairing Hamiltonians. Fidelity of the target states corresponding to each algorithm is numerically …
For variational algorithms on the near term quantum computing hardware, it is highly desirable to use very accurate ansatze with low implementation cost. Recent studies have shown that the antisymmetrized geminal power (AGP) wavefunction can be an excellent starting point for ansatze describing systems with strong pairing correlations, as those occurring in superconductors. In this work, we show how AGP can be efficiently implemented on a quantum computer with circuit depth, number of CNOTs, and number of measurements being linear in system size. Using AGP as the initial reference, we propose …
An efficient quantum algorithm is proposed to solve in polynomial time the parity problem, one of the hardest problems both in conventional quantum computation and in classical computation, on NMR quantum computers. It is based on the quantum parallelism principle in a quantum ensemble, the selective decoherence manipulation, and the NMR phase-sensitive measurement. The quantum circuit for the quantum algorithm is designed explicitly.
We define formally decohered quantum computers (using density matrices), and present a simulation of them by a probabalistic classical Turing Machine. We study the slowdown of the simulation for two cases: (1) sequential quantum computers, or quantum Turing machines(QTM), and (2) parallel quantum computers, or quantum circuits. This paper shows that the computational power of decohered quantum computers depends strongly on the amount of parallelism in the computation. The expected slowdown of the simulation of a QTM is polynomial in time and space of the quantum computation, for any non zero d…
An integer $a$ is a quadratic nonresidue for a prime $p$ if $x^2 \equiv a \bmod p$ has no solution. Quadratic nonresidues may be found by probabilistic methods in polynomial time. However, without assuming the Generalized Riemann Hypothesis, no deterministic polynomial-time algorithm is known. We present a quantum algorithm which generates a random quadratic nonresidue in deterministic polynomial time.
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical …
We present a program for solving exactly the general pairing Hamiltonian based on diagonalization. The program generates the seniority-zero shell-model-like basis vectors via the `01' inversion algorithm. The Hamiltonian matrix is constructed in that seniority-zero space. The program evaluates all non-zero elements of the Hamiltonian matrix "on the fly" using the scattering operator and the search algorithm that act on the generated basis. The matrix is diagonalized by using the iterative Lanczos algorithm. The program thus developed, PairDiag, can calculate efficiently the ground-state eigenv…
Quantum simulation uses a well-known quantum system to predict the behavior of another quantum system. Certain limitations in this technique arise, however, when applied to specific problems, as we demonstrate with a theoretical and experimental study of an algorithm to find the low-lying spectrum of a Hamiltonian. While the number of elementary quantum gates does scale polynomially with the size of the system, it increases inversely to the desired error bound $ε$. Making such simulations robust to decoherence using fault-tolerance constructs requires an additional factor of $1/ ε$ gates. Thes…
We investigate the power of quantum computers when they are required to return an answer that is guaranteed correct after a time that is upper-bounded by a polynomial in the worst case. In an oracle setting, it is shown that such machines can solve problems that would take exponential time on any classical bounded-error probabilistic computer.
We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its poly…
We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional post-processing. The final goal is to develop configuration interaction techniques based on many-body trial states pre-optimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and quantum Krylov approaches for the post-processing. The latter metho…
The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can only be applied to small systems. By contrast, we demonstrate that quantum computers could exactly simulate chemical reactions in polynomial time. Our algorithm uses the split-operator approach and explicitly simulates all electron-nuclear and inter-electronic interactions in quadratic time. Surprisingly, this treatment is not only more accurate than the Born-Oppenheimer approximation, but faster and more efficient as well, for al…
It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable problems. Here we show that evaluating a certain approximation to the Jones polynomial at a fifth root of unit…
Starting from the Quantum-Phase-Estimate (QPE) algorithm, a method is proposed to construct entangled states that describe correlated many-body systems on quantum computers. Using operators for which the discrete set of eigenvalues is known, the QPE approach is followed by measurements that serve as projectors on the entangled states. These states can then be used as inputs for further quantum or hybrid quantum-classical processing. When the operator is associated to a symmetry of the Hamiltonian, the approach can be seen as a quantum--computer formulation of symmetry breaking followed by symm…
The quantum search problem is an important problem due to the fact that a general NP problem can be solved efficiently by an unsorted quantum search algorithm. Here it has been shown that the quantum search problem could be solved in polynomial time on an NMR quantum computer. The NMR ensemble quantum computation is based on the quantum mechanical unitary dynamics that both a closed quantum system and its ensemble obey the same quantum mechanical unitary dynamics instead of on the pseudopure state or the effective pure state of the classical NMR quantum computation. Based on the new principle …
Recent works have shown that quantum computers can polynomially speed up certain SAT-solving algorithms even when the number of available qubits is significantly smaller than the number of variables. Here we generalise this approach. We present a framework for hybrid quantum-classical algorithms which utilise quantum computers significantly smaller than the problem size. Given an arbitrarily small ratio of the quantum computer to the instance size, we achieve polynomial speedups for classical divide-and-conquer algorithms, provided that certain criteria on the time- and space-efficiency are me…
We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible cyclic codes. This problem is related to the evaluation of the Jones and Tutte polynomials. We consider the connection between the weight enumerator polynomial from coding theory and Z and exploit the fact that there exists a quantum algorithm for efficiently estimating Gauss sums in order to obtain the weight enumerator for a certain class of linear codes. In this way we demonstrate that for …
Simulating of exotic phases of matter that are not amenable to classical techniques is one of the most important potential applications of quantum information processing. We present an efficient algorithm for preparing a large class of topological quantum states -- the G-injective Projected Entangled Pair States (PEPS) -- on a quantum computer. Important examples include the resonant valence bond (RVB) states, conjectured to be topological spin liquids. The runtime of the algorithm scales polynomially with the condition number of the PEPS projectors, and inverse-polynomially in the spectral ga…
The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variab…
The energy gap is calculated for the ground state quantum computer circuit, which was recently proposed by Mizel et.al. When implementing a quantum algorithm by Hamiltonians containing only pairwise interaction, the inverse of energy gap $1/Δ$ is proportional to $N^{4k}$, where $N$ is the number of bits involved in the problem, and $N^k$ is the number of control operations performed in a standard quantum paradigm. Besides suppressing decoherence due to the energy gap, in polynomial time ground state quantum computer can finish the quantum algorithms that are supposed to be implemented by stand…
This expository paper reviews some of the recent uses of computational algebraic geometry in classical and quantum optimization. The paper assumes an elementary background in algebraic geometry and adiabatic quantum computing (AQC), and concentrates on presenting concrete examples (with Python codes tested on a quantum computer) of applying algebraic geometry constructs: solving binary optimization, factoring, and compiling. Reversing the direction, we also briefly describe a novel use of quantum computers to compute Groebner bases for toric ideals. We also show how Groebner bases play a role …
We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat's method in classical computing. All these algorithms are polynomial in the input size.
We present a quantum probabilistic algorithm which tests with a polynomial computational complexity whether a given composite number is of the Carmichael type. We also suggest a quantum algorithm which could verify a conjecture by Pomerance, Selfridge and Wagstaff concerning the asymptotic distribution of Carmichael numbers smaller than a given integer.
We study the properties of eigenstates of an operating quantum computer which simulates the dynamical evolution in the regime of quantum chaos. Even if the quantum algorithm is polynomial in number of qubits $n_q$, it is shown that the ideal eigenstates become mixed and strongly modified by static imperfections above a certain threshold which drops exponentially with $n_q$. Above this threshold the quantum eigenstate entropy grows linearly with $n_q$ but the computation remains reliable during a time scale which is polynomial in the imperfection strength and in $n_q$.
A subtle relation between Quantum Hall physics and the phenomenon of pairing is unveiled. By use of second quantization, we establish a connection between (i) a broad class of rotationally symmetric two-body interactions within the lowest Landau level and (ii) integrable hyperbolic Richardson-Gaudin type Hamiltonians that arise in (p_{x}+ip_{y}) superconductivity. Specifically, we show that general Haldane pseudopotentials (and their sums) can be expressed as a sum of repulsive non-commuting (p_{x}+ip_{y})-type pairing Hamiltonians. For the Laughlin sequence, it is observed that this problem i…
An important problem in quantum information is the practical demonstration of non-classical long-range order on quantum computers. One of the best known examples of a quantum system with non-classical long-range order is a superconductor. Here we achieve Cooper pairing of qubits on a quantum computer to represent superconducting or superfluid states. We rigorously confirm the quantum long-range order by measuring the large $O(N)$ eigenvalue of the two-electron reduced density matrix. The demonstration of maximal quantum long-range order is an important step towards more complex modeling of sup…
A line of work initiated by Terhal and DiVincenzo and Bremner, Jozsa, and Shepherd, shows that quantum computers can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. Aaronson and Arkhipov take this further by considering a distribution that can be sampled efficiently by linear optical quantum computation, that under two feasible conjectures, cannot even be approximately sampled classically within bounded total variation distance, unless the PH collapses. In this work we use Quantum Fourier Sampling to…
Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibo…
