We describe methods for simulating general second-quantized Hamiltonians using the compact encoding, in which qubit states encode only the occupied modes in physical occupation number basis states. These methods apply to second-quantized Hamiltonians composed of a constant number of interactions, i.e., linear combinations of ladder operator monomials of fixed form. Compact encoding leads to qubit requirements that are optimal up to logarithmic factors. We show how to use sparse Hamiltonian simulation methods for second-quantized Hamiltonians in compact encoding, give explicit implementations for the required oracles, and analyze the methods. We also describe several example applications including the free boson and fermion theories, the $ϕ^4$-theory, and the massive Yukawa model, all in both equal-time and light-front quantization. Our methods provide a general-purpose tool for simulating second-quantized Hamiltonians, with optimal or near-optimal scaling with error and model parameters.
We present a method for encoding second-quantized fermionic systems in qubits when the number of fermions is conserved, as in the electronic structure problem. When the number $F$ of fermions is much smaller than the number $M$ of modes, this symmetry reduces the number of information-theoretically required qubits from $Θ(M)$ to $O(F\log M)$. In this limit, our encoding requires $O(F^2\log^4 M)$ qubits, while encoded fermionic creation and annihilation operators have cost $O(F^2\log^5 M)$ in two-qubit gates. When incorporated into randomized simulation methods, this permits simulating time-evo…
In this dissertation a simple Hamiltonian for a system of inter-acting molecules and radiation field is developed from a model of N Two-Level Molecules interacting, via a dipole approximation, with a single mode, quantized radiation field. The exact eigenvalues and eigenvectors of this Hamiltonian are compared with analytical results obtained from several different approximation schemes applied to the simple model. The consequence of this comparison is the development of validity criteria for the utilization of these approximation schemes to more difficult problems. For example, an analysis do…
The quantum singular value transformation is a powerful quantum algorithm that allows one to apply a polynomial transformation to the singular values of a matrix that is embedded as a block of a unitary transformation. This paper shows how to perform the quantum singular value transformation for a matrix that can be embedded as a block of a Hamiltonian. The transformation can be implemented in a purely Hamiltonian context by the alternating application of Hamiltonians for chosen intervals: it is an example of the Quantum Alternating Operator Ansatz (generalized QAOA). We also show how to use t…
We discuss encodings of fermionic many-body systems by qubits in the presence of symmetries. Such encodings eliminate redundant degrees of freedom in a way that preserves a simple structure of the system Hamiltonian enabling quantum simulations with fewer qubits. First we consider $U(1)$ symmetry describing the particle number conservation. Using a previously known encoding based on the first quantization method a system of $M$ fermi modes with $N$ particles can be simulated on a quantum computer with $Q=N\log{(M)}$ qubits. We propose a new version of this encoding tailored to variational quan…
Block-Invariant Symmetry Shift: Preprocessing technique for second-quantized Hamiltonians to improve their decompositions to Linear Combination of Unitaries
Noncontextual Pauli Hamiltonians decompose into sets of Pauli terms to which joint values may be assigned without contradiction. We construct a quasi-quantized model for noncontextual Pauli Hamiltonians. Using this model, we give an algorithm to classically simulate noncontextual VQE. We also use the model to show that the noncontextual Hamiltonian problem is NP-complete. Finally, we explore the applicability of our quasi-quantized model as an approximate simulation tool for contextual Hamiltonians. These results support the notion of noncontextuality as classicality in near-term quantum algor…
We present a novel approach for studying the dynamics of a superconducting qubit in a cavity. We succeed in linearizing the Hamiltonian through the application of an appropriate unitary transformation followed by a rotating wave approximation (RWA). For certain values of the parameters involved, we show that it is possible to obtain a a Jaynes-Cummings type Hamiltonian. As an example, we show the existence of super-revivals for the qubit inversion.
Low-gate-count block encodings for second-quantized fermionic Hamiltonians
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of Minnesota（明尼苏达大学）；Duke University（杜克大学）；Georgetown University（乔治城大学）；University of California, Berkeley（加州大学伯克利分校）；Google Quantum AI（谷歌量子AI）
We show that the interaction between a movable mirror with a quantized field that interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operators (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a series of small rotations, into an effective Hamiltonian that may be solved. We would like to stress that the transformation in terms of SGO already simplifies enough the Hamiltonian in the sense that, in an exact way, it "eliminates" one of the three-subsystems, namely the quantized field.
We construct a non-perturbative approach based on quantum averaging combined with resonant transformations to detect the resonances of a given Hamiltonian and to treat them. This approach, that generalizes the rotating-wave approximation, takes into account the resonances at low field and also at high field (non-linear resonances). This allows to derive effective Hamiltonians that contain the qualitative features of the spectrum, i.e. crossings and avoided crossings, as a function of the coupling constant. At a second stage the precision of the spectrum can be improved quantitatively by standa…
In this work we present a method for generating a fermionic encoding tailored to a set of target fermionic operators and to a target hardware connectivity. Our method uses brute force search, over the space of all encodings which map from Majorana monomials to Pauli operators, to find an encoding which optimizes a target cost function. In contrast to earlier works in this direction, our method searches over an extremely broad class of encodings which subsumes all known second quantized encodings that constitute algebra homomorphisms. In order to search over this class, we give a clear mathemat…
We present an encoding technique that reduces the effects of noise on quantum spin systems whose operation is driven by Hamiltonian evolution. This technique is widely applicable, being most relevant to the scenarios where there are insufficient qubits to permit full scale error correction. Instead, our technique can be implemented over small numbers of qubits and still leads to noticeable improvements in the fidelity of operations. The encoding scheme is easy to implement, flexible with respect to choice of Hamiltonian, and close to optimal.
Recently there have been some controversies about the criterion of the adiabatic approximation. It is shown that an approximate diagonalization of the effective Hamiltonian in the second quantized formulation gives rise to a reliable and unambiguous criterion of the adiabatic approximation. This is illustrated for the model of Marzlin and Sanders and a model related to the geometric phase which can be exactly diagonalized in the present sense.
Efficient Lattice Hamiltonian Encoding for the Shortest Vector Problem
Imperial College London（帝国理工学院）；Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University（特拉维夫大学雷蒙德和贝弗利萨克勒物理与天文学学院）；Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿罗森多夫亥姆霍兹中心）
Due to the limitations of present-day quantum hardware, it is especially critical to design algorithms that make the best possible use of available resources. When simulating quantum many-body systems on a quantum computer, straightforward encodings that transform many-body Hamiltonians into qubit Hamiltonians use $N$ of the available basis states of an $N$-qubit system, whereas $2^N$ are in theory available. We explore an efficient encoding that uses the entire set of basis states, where terms in the Hamiltonian are mapped to qubit operators with a Hamiltonian that acts on the basis states in…
The Born-Fock theorem is one of the most fundamental theorems of quantum mechanics and forms the basis for reliable and efficient navigation in the Hilbert space of a quantum system with a time-dependent Hamiltonian by adiabatic evolution. In the absence of level crossings, i.e. without degeneracies, and under adiabatic time evolution all eigenstates of the Hamiltonian keep their energetic order, labelled by a conserved integer quantum number. Thus controlling the eigenstates of the Hamiltonian and their energetic order in asymptotic limits allows to engineer a perfect adiabatic transfer betwe…
We examine the analytical structure of the nonlinear Lienard oscillator and show that it is a bi-Hamiltonian system depending upon the choice of the coupling parameters. While one has been recently studied in the context of a quantized momentum- dependent mass system, the other Hamiltonian also reflects a similar feature in the mass function and also depicts an isotonic character. We solve for such a Hamiltonian and give the complete solution in terms of a confluent hypergeometric function.
Second-quantized approach to the study of Halperin state in fractional quantum Hall effect
College of Physics and Electronic Science, Hubei Normal University（湖北师范大学物理与电子科学学院）；Hubei Key Laboratory of Photoelectric Materials and Devices, Hubei Normal University（湖北师范大学光电材料与器件湖北省重点实验室）
A semiclassical (SC) approach is developed for nonequilibrium quantum transport in molecular junctions. Following the early work of Miller and White [J. Chem. Phys. 84, 5059 (1986)], the many-electron Hamiltonian in second quantization is mapped onto a classical model that preserves the fermionic character of electrons. The resulting classical electronic Hamiltonian allows for real-time molecular dynamics simulations of the many-body problem from an uncorrelated initial state to the steady state. Comparisons with exact results generated for the resonant level model reveal that a semiclassical …
We propose a computational protocol for quantum simulations of Fermionic Hamiltonians on a quantum computer, enabling calculations which were previously not feasible with conventional encoding and ansatses of variational quantum eigensolvers (VQE). We combine a qubit-efficient encoding scheme mapping Slater determinants onto qubits with a modified qubit-coupled cluster ansatz and noise-mitigation techniques. Our strategy leads to a substantial improvement in the scaling of circuit gate counts and to a decrease in the number of required variational parameters, thus increasing the resilience to …
We present quantum algorithms for the estimation of n-time correlation functions, the local and non-local density of states, and dynamical linear response functions. These algorithms are all based on block-encodings - a versatile technique for the manipulation of arbitrary non-unitary matrices on a quantum computer. We describe how to 'sketch' these quantities via the kernel polynomial method which is a standard strategy in numerical condensed matter physics. These algorithms use amplitude estimation to obtain a quadratic speedup in the accuracy over previous results, can capture any observabl…
Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum (NISQ) computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We pro…
The dispersive interaction between a two-level atom and a quantized field is studied. We consider besides a time dependent linear amplification and dissipative processes. In order to solve the master equation for this system, we use superoperator techniques.
Simulation of quantum systems is expected to be one of the most important applications of quantum computing, with much of the theoretical work so far having focused on fermionic and spin-$\frac{1}{2}$ systems. Here, we instead consider encodings of $d$-level (i.e. qudit) quantum operators into multi-qubit operators, studying resource requirements for approximating operator exponentials by Trotterization. We primarily focus on spin-$s$ and truncated bosonic operators in second quantization, observing desirable properties for approaches based on the Gray code, which to our knowledge has not been…
We show that, by means of a right-unitary transformation, the fully quantized Landau-Zener Hamiltonian in the weak-coupling regime may be solved by using known solutions from the standard Landau-Zener problem. In the strong-coupling regime, where the rotating wave approximation is not valid, we show that the quantized Landau-Zener Hamiltonian may be diagonalized in the atomic basis by means of a unitary transformation; hence allowing numerical solutions for the few photons regime via truncation.
We give here a field-theoretical derivation of the Hamiltonian of the non-relativistic quantum electrodynamics in the Coulomb gauge using the Lagrange formalism. It leads to the same result as the usual derivation, where one just replaces the classical vector potential in the minimal coupling of the second quantized electron Hamiltonian by the quantized one and adds the photon energy. This approach illustrates however the proper use of the Euler-Lagrange equations and the canonical formalism that fail if one tries to build up the many-body quantum theory starting from the classical theory of p…
We study an ultracold gas of neutral atoms subject to the periodic optical potential generated by a high-$Q$ cavity mode. In the limit of very low temperatures, cavity field and atomic dynamics require a quantum description. Starting from a cavity QED single atom Hamiltonian we use different routes to derive approximative multiparticle Hamiltonians in Bose-Hubbard form with rescaled or even dynamical parameters. In the limit of large enough cavity damping the different models agree. Compared to free space optical lattices, quantum uncertainties of the potential and the possibility of atom-fiel…
We investigate a quantum parametric amplifier with dynamical pump mode, viewed as a zero-dimensional model of Hawking radiation from an evaporating black hole. The conditions are derived under which the spectrum of particles generated from vacuum fluctuations deviates from the thermal spectrum predicted for the conventional parametric amplifier. We find that significant deviations arise when the pump mode (black hole) has emitted nearly half of its initial energy into the signal (Hawking radiation) and idler (in-falling particle) modes. As a model of black hole dynamics, this finding lends sup…
Multiple quantum coherences are typically characterised by their coherence number and the number of spins that make up the state, though only the coherence number is normally measured. We present a simple set of measurements that extend our knowledge of the multiple quantum state by recording the coherences in both the $x$ basis and the usual $z$ basis. The coherences in the two bases are related by a similarity transformation. We characterize the growth of the multiple quantum coherences via measurements in the two bases, and show that the rate varies with the coefficient of the driving term …
Simulation of fermionic many-body systems on a quantum computer requires a suitable encoding of fermionic degrees of freedom into qubits. Here we revisit the Superfast Encoding introduced by Kitaev and one of the authors. This encoding maps a target fermionic Hamiltonian with two-body interactions on a graph of degree $d$ to a qubit simulator Hamiltonian composed of Pauli operators of weight $O(d)$. A system of $m$ fermi modes gets mapped to $n=O(md)$ qubits. We propose Generalized Superfast Encodings (GSE) which require the same number of qubits as the original one but have more favorable pro…
Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of …
It is proposed to map the quantum information qubit not to individual spin 1/2 states, but to the collective spin states being eigenfunctions of the Hamiltonian including spin-spin interactions, which may be not small. Such an approach allows to introduce more stable calculation basis for quantum computer based on the solid state NMR systems.
