Quantum impurity models using superpositions of fermionic Gaussian states: Practical methods and applications

formula omitted]QIST v0.7: An open source continuous-time quantum Monte Carlo impurity solver toolkit

We introduce an efficient method to quantify nonstabilizerness in fermionic Gaussian states, overcoming the long-standing challenge posed by their extensive entanglement. Using a perfect sampling scheme based on an underlying determinantal point process, we compute the Stabilizer Renyi Entropies (SREs) for systems with hundreds of qubits. Benchmarking on random Gaussian states with and without particle conservation, we reveal an extensive leading behavior equal to that of Haar random states, with logarithmic subleading corrections. We support these findings with analytical calculations for a set of related quantities, the participation entropies in the computational (or Fock) basis, for which we derive an exact formula. We also investigate the time evolution of non-stabilizerness in a random unitary circuit with Gaussian gates, observing that it converges in a time that scales logarithmically with the system size. Applying the sampling algorithm to a two-dimensional free-fermionic topological model, we uncover a sharp transition in non-stabilizerness at the phase boundaries, highlighting the power of our approach in exploring different phases of quantum many-body systems, even in higher dimensions.

A recovery map effectively cancels the action of a quantum operation to a partial or full extent. We study the Petz recovery map in the case where the quantum channel and input states are fermionic and Gaussian. Gaussian states are convenient because they are totally determined by their covariance matrix and because they form a closed set under so-called Gaussian channels. Using a Grassmann representation of fermionic Gaussian maps, we show that the Petz recovery map is also Gaussian and determine it explicitly in terms of the covariance matrix of the reference state and the data of the channel. As a by-product, we obtain a formula for the fidelity between two fermionic Gaussian states. We also discuss subtleties arising from the singularities of the involved matrices.

Quantum information and quantum computing offer a new range of possibilities in terms ofnew computing algorithms, the ability to perform new types of operations and make statementsabout information in terms of what it is physically possible to know about a system.Experiments in quantum information and quantum computing have been advancing significantly,especially over the last decade. In optical quantum computing there are two maintypes of experiments that are performed today. They are experiments involving single photonswith information encoded on the polarisation of the photon and experiments involvingstates with Gaussian noise profiles in their quadratures. There appears to be no way toperform the required operations for universal quantum computing using operations whichtake Gaussian states to Gaussian states. Universal quantum computing using single photonsis possible but uses either enormously large non-linearities or a large number of resources.Also, operations which take Gaussian states to non-Gaussian states are not well understoodin general and can often be very difficult to implement.This thesis presents three sources of optical non-Gaussian states which demonstrate someof the interesting properties of these states. First, a scheme for universal quantum computingusing superpositions of coherent states is presented. Error correction and the fault toleranceproperties of this scheme is also explored and compared with current single photon architectures.These gates require superpositions of coherent states as resources and a relativelysimple scheme is shown for how these states may be generated. Next a scheme for distillingnear ideal Gaussian entanglement using non-Gaussian operations is presented. The loss andnoise conditions under which ideal distillation can be achieved are presented. Finally, a verysimple way of generating mixed non-Gaussian states from two Gaussian states is presented.These states are shown to give rise to counter-intuitive results when measuring their entanglementand when analysing their operational ability in terms of teleportation and quantumkey distribution.

We derive stochastic master equations for a quantum system interacting with a Bose field prepared in a superposition of continuous-mode coherent states. To determine a conditional evolution of the quantum system, we use a collision model with an environment given as an infinite chain of not interacting between themselves qubits prepared initially in an entangled state being a discrete analogue of a superposition of coherent states of the Bose field. The elements of the environment chain interact with the quantum system in turn one by one, and they are subsequently measured. We determine a conditional evolution of the quantum system for continuous in time observations of the output field as a limit of discrete recurrence equations. We consider the stochastic master equations for a counting as well as for a diffusive stochastic process.

In this paper, a new state called superposition of three-mode photon-added trio coherent state is introduced and its nonclassical properties are studied. Based on the negative values of the Wigner function, we find that the nonclassical as well as non-Gaussian characteristics of the superposition of three-mode photon-added trio coherent state are more obvious than that of the original trio coherent state. The superposition of three-mode photon-added trio coherent state always appears three-mode sum squeezing while the trio coherent state does not. It also shows that the superposition of three-mode photon-added trio coherent state is genuinely entangled. The entanglement degree of this state is more improved compared with the trio coherent state. It becomes bigger and bigger through nonlocally increasing the number of added photons.

We calculate the second-order moments, the Robertson-Schrödinger uncertainty product, and the Mandel factor for various superpositions of coherent phase states with opposite arguments, comparing the results with similar superpositions of the usual (Klauder-Glauber-Sudarshan) coherent states. We discover that the coordinate variance in the analog of even coherent states can show the most strong squeezing effect, close to the maximal possible squeezing for the given mean photon number. On the other hand, the Robertson-Schrödinger (RS) uncertainty product in superpositions of coherent phase states increases much slower (as function of the mean photon number) than in superpositions of the usual coherent states. A nontrivial behavior of the Mandel factor for small mean photon numbers is discovered in superpositions with unequal weights of two components. An exceptional nature of the even and odd superpositions is demonstrated.

Explicit Pfaffian formula for amplitudes of fermionic Gaussian pure states in arbitrary Pauli bases

“Classical shadows” are estimators of an unknown quantum state, constructed from suitably distributed random measurements on copies of that state (Huang et al. in Nat Phys 16:1050, 2020, https://doi.org/10.1038/s41567-020-0932-7). In this paper, we analyze classical shadows obtained using random matchgate circuits, which correspond to fermionic Gaussian unitaries. We prove that the first three moments of the Haar distribution over the continuous group of matchgate circuits are equal to those of the discrete uniform distribution over only the matchgate circuits that are also Clifford unitaries; thus, the latter forms a “matchgate 3-design.” This implies that the classical shadows resulting from the two ensembles are functionally equivalent. We show how one can use these matchgate shadows to efficiently estimate inner products between an arbitrary quantum state and fermionic Gaussian states, as well as the expectation values of local fermionic operators and various other quantities, thus surpassing the capabilities of prior work. As a concrete application, this enables us to apply wavefunction constraints that control the fermion sign problem in the quantum-classical auxiliary-field quantum Monte Carlo algorithm (QC-AFQMC) (Huggins et al. in Nature 603:416, 2022, https://doi.org/10.1038/s41586-021-04351-z), without the exponential post-processing cost incurred by the original approach.

In unitary circuit games, two competing parties, an "entangler" and a "disentangler", can induce an entanglement phase transition in a quantum many-body system. The transition occurs at a certain rate at which the disentangler acts. We analyze such games within the context of matchgate dynamics, which equivalently corresponds to evolutions of non-interacting fermions. We first investigate general entanglement properties of fermionic Gaussian states (FGS). We introduce a representation of FGS using a minimal matchgate circuit capable of preparing the state and derive an algorithm based on a generalized Yang-Baxter relation for updating this representation as unitary operations are applied. This representation enables us to define a natural disentangling procedure that reduces the number of gates in the circuit, thereby decreasing the entanglement contained in the system. We then explore different strategies to disentangle the systems and study the unitary circuit game in two different scenarios: with braiding gates, i.e., the intersection of Clifford gates and matchgates, and with generic matchgates. For each model, we observe qualitatively different entanglement transitions, which we characterize both numerically and analytically.

We exploit insights into the geometry of bosonic and fermionic Gaussian states to develop an efficient local optimization algorithm to extremize arbitrary functions on these families of states. The method is based on notions of gradient descent attuned to the local geometry which also allows for the implementation of local constraints. The natural group action of the symplectic and orthogonal group enables us to compute the geometric gradient efficiently. While our parametrization of states is based on covariance matrices and linear complex structures, we provide compact formulas to easily convert from and to other parametrization of Gaussian states, such as wave functions for pure Gaussian states, quasiprobability distributions and Bogoliubov transformations. We review applications ranging from approximating ground states to computing circuit complexity and the entanglement of purification that have both been employed in the context of holography. Finally, we use the presented methods to collect numerical and analytical evidence for the conjecture that Gaussian purifications are sufficient to compute the entanglement of purification of arbitrary mixed Gaussian states.

Out-of-equilibrium fermionic quantum impurity models (QIM), describing a small interacting system coupled to a continuous fermionic bath, play an important role in condensed matter physics. Solving such models is a computationally demanding task, and a variety of computational approaches are based on finding approximate representations of the bath by a finite number of modes. In this paper, we formulate the problem of finding efficient bath representations as that of approximating a kernel of the bath's Feynman-Vernon influence functional by a sum of complex exponentials, with each term defining a fermionic pseudomode. Under mild assumptions on the analytic properties of the bath spectral density, we provide an analytic construction of pseudomodes, and prove that their number scales polylogarithmically with the maximum evolution time $T$ and the approximation error $\ensuremath{\varepsilon}$. We then demonstrate that the number of pseudomodes can be significantly reduced by an interpolative matrix decomposition (ID). Furthermore, we present a complementary approach, based on constructing rational approximations of the bath's spectral density using the ``AAA'' algorithm, followed by compression with ID. The combination of two approaches yields a pseudomode count scaling as ${N}_{\text{ID}}\ensuremath{\sim}ln(T)ln(1/\ensuremath{\varepsilon})$, and the agreement between the two approaches suggests that the result is close to optimal. Finally, to relate our findings to QIM, we derive an explicit Liouvillian that describes the time evolution of the combined impurity-pseudomodes system. These results establish bounds on the computational resources required for solving out-of-equilibrium QIMs, providing an efficient starting point for tensor-network methods for QIMs.

As a specific trait of quantum mechanics, quantum superposition is exploited in many applications such as quantum computation. This paper proposes a feedback scheme to stabilize the desired quantum superposition states. To this aim, a new factorization is derived for the stochastic master equation of a quantum system which is driven by the field in the superposition of coherent states. Then, a feedback scheme is proposed, which consists of a quantum system driven by the field in the superposition of coherent states and a classical feedback channel proportional to the observations of a Homodyne detector. Afterward, the quantum filter of the closed loop system is attained using the SLH framework. Finally, the factorization of the closed loop quantum filter is obtained and the control law is calculated such that the proposed factorization satisfies the stabilization condition. Finally, as an example, the control method is simulated for a two-level atom and the stabilization of the superposition states is achieved.

Electronic states in low-dimensional nano-structures: Comparison between the variational and plane wave basis method

Role of quantum interference in the origin of higher-order nonclassical characteristics of radiation field has been probed vis-a-vis a discrete and a continuous superposition of coherent states. Specifically, the possibilities of observing higher-order nonclassical properties (e.g., higher-order antibunching (HOA), higher-order sub-Poissonian photon statistics (HOSPS), higher-order squeezing (HOS) of Hong-Mandel type and Hillery type) have been investigated using a shifted symmetric cat state that reduces to Yurke-Stoler, even and odd coherent states at various limits. This shifted symmetric cat state which can be viewed as a discrete superposition of coherent states is found to show HOA and HOSPS. Similarly, higher-order nonclassical properties of a one-dimensional continuous superposition of coherent states is also studied here. The investigation has revealed the existence of HOS and HOSPS in the one-dimensional continuous superposition of coherent states studied here. Effect of non-Gaussianity inducing operations (e.g., photon addition and addition followed by subtraction) on these superposition states have also been investigated. Finally, some comparisons have been made between the higher-order nonclassical properties of discrete and continuous superposition of coherent states.