Dimensional Hilbert Space Research Articles

Experimental full calibration of quantum devices in a semi-device-independent way

Semi-device-independent (SDI) methods offer a credible way to calibrate preparation and measurement devices simultaneously in quantum information processing, using only prior knowledge such as the Hilbert space dimension. To date, the SDI method is restricted to a few state paradigms, which impedes its broader applications. Recently, Tavakoli [Phys. Rev. Lett. 125, 150503 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.150503] proposed an SDI scheme to certify t-designs with discrete and symmetric structures. In this work, we bridge the gap between discrete and continuous structures with a concept termed “covering angle,” while maintaining the SDI feature. This concept enables us to evaluate a quantum device’s ability to generate arbitrary quantum states in a Hilbert space via calibrating a certain t-design. This so-called full calibration method is further tailored to be tolerant of errors in realistic state production. We demonstrate this full calibration scheme for a qubit system with various t-designs and show that it renders SDI certificates for quantum key distribution, quantum random number generation, and magic state distillability.

SVD entanglement entropy

In this paper, we introduce a new quantity called SVD entanglement entropy. This is a generalization of entanglement entropy in that it depends on two different states, as in pre- and post-selection processes. This SVD entanglement entropy takes non-negative real values and is bounded by the logarithm of the Hilbert space dimensions. The SVD entanglement entropy can be interpreted as the average number of Bell pairs distillable from intermediates states. We observe that the SVD entanglement entropy gets enhanced when the two states are in the different quantum phases in an explicit example of the transverse-field Ising model. Moreover, we calculate the Rényi SVD entropy in various field theories and examine holographic calculations using the AdS/CFT correspondence.

Simulation of quantum optics by coherent state decomposition

We introduce a framework for simulating quantum optics by decomposing the system into a finite rank (number of terms) superposition of coherent states. This allows us to define a resource theory, where linear optical operations are “free” (i.e., do not increase the rank), and the simulation complexity for an m-mode system scales quadratically in m, in stark contrast to the Hilbert space dimension. We outline this approach explicitly in the Fock basis, relevant in particular for Boson sampling, where the simulation time (space) complexity for computing output amplitudes, to arbitrary accuracy, scales as O(m2 2 n ) [O(m2 n )] for n photons distributed among m modes. We additionally demonstrate that linear optical simulations with the n photons initially in the same mode scales efficiently, as O(m2 n). This paradigm provides a practical notion of “non-classicality,” i.e., the classical resources required for simulation. Moreover, by making connections to the stellar rank formalism, we show this comes from two independent contributions, the number of single-photon additions and the amount of squeezing.

Dimensionality Reduction with Variational Encoders Based on Subsystem Purification

Efficient methods for encoding and compression are likely to pave the way toward the problem of efficient trainability on higher-dimensional Hilbert spaces, overcoming issues of barren plateaus. Here, we propose an alternative approach to variational autoencoders to reduce the dimensionality of states represented in higher dimensional Hilbert spaces. To this end, we build a variational algorithm-based autoencoder circuit that takes as input a dataset and optimizes the parameters of a Parameterized Quantum Circuit (PQC) ansatz to produce an output state that can be represented as a tensor product of two subsystems by minimizing Tr(ρ2). The output of this circuit is passed through a series of controlled swap gates and measurements to output a state with half the number of qubits while retaining the features of the starting state in the same spirit as any dimension-reduction technique used in classical algorithms. The output obtained is used for supervised learning to guarantee the working of the encoding procedure thus developed. We make use of the Bars and Stripes (BAS) dataset for an 8 × 8 grid to create efficient encoding states and report a classification accuracy of 95% on the same. Thus, the demonstrated example provides proof for the working of the method in reducing states represented in large Hilbert spaces while maintaining the features required for any further machine learning algorithm that follows.

Quantum detection problem for fusion frames

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.

Understanding the mapping of encode data through an implementation of quantum topological analysis

A potential advantage of quantum machine learning stems from the ability of encoding classical data into high dimensional complex Hilbert space using quantum circuits. Recent studies exhibit that not all encoding methods are the same when representing classical data since certain parameterized circuit structures are more expressive than the others. In this study, we show the difference in encoding techniques can be visualized by investigating the topology of the data embedded in complex Hilbert space. The technique for visualization is a hybrid quantum based topological analysis which uses simple diagonalization of the boundary operators to compute the persistent Betti numbers and the persistent homology graph. To augment the computation of Betti numbers within a NISQ framework, we suggest a simple hybrid algorithm. Through an illuminating example of a synthetic data set and the methods of angle encoding, amplitude encoding, and IQP encoding, we reveal topological differences between the encoding methods, as well as differences with the original data. Consequently, our results suggest the encoding method needs to be considered carefully within different quantum machine learning models since it can strongly affect downstream analysis like clustering or classification.

Subproduct systems with quantum group symmetry

We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones–Wenzl projections in Temperley–Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C ^* -algebras that are KK -equivalent to \mathbb C and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz–Pimsner algebras under monoidal equivalence of quantum symmetry groups.

Quantitative bounds for unconditional pairs of frames

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [22]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all C>0 and N∈N the following is true: Let (xj)j=1N and (fj)j=1N be sequences in a finite dimensional Hilbert space which satisfy ‖xj‖=‖fj‖ for all 1≤j≤N and‖∑j=1Nεj〈x,fj〉xj‖≤C‖x‖,for all x∈ℓ2M and |εj|=1. If the frame operator for (fj)j=1N has eigenvalues λ1≥...≥λM and β>0 is such that λ1≤βM−1∑j=1Mλj then (fj)j=1N has Bessel bound 27β2C. The same holds for (xj)j=1N.

The semi-Fredholmness and property (ω) for 2×2 upper triangular operator matrices

Let M C = ( A C 0 B ) be a 2 × 2 upper triangular operator matrix acting on H ⊕ K , where H and K are complex infinite dimensional separable Hilbert spaces. In this paper, we study the semi-Fredholmness for M C for some invertible C ∈ B ( K , H ) , and using the conclusions obtained, we characterize the equivalent conditions such that M C ∈ ( ω ) for any invertible C ∈ B ( K , H ) .

Quantum Gauge Networks: A New Kind of Tensor Network

Although tensor networks are powerful tools for simulating low-dimensional quantum physics, tensor network algorithms are very computationally costly in higher spatial dimensions. We introduce quantum gauge networks: a different kind of tensor network ansatz for which the computation cost of simulations does not explicitly increase for larger spatial dimensions. We take inspiration from the gauge picture of quantum dynamics, which consists of a local wavefunction for each patch of space, with neighboring patches related by unitary connections. A quantum gauge network (QGN) has a similar structure, except the Hilbert space dimensions of the local wavefunctions and connections are truncated. We describe how a QGN can be obtained from a generic wavefunction or matrix product state (MPS). All 2k-point correlation functions of any wavefunction for M many operators can be encoded exactly by a QGN with bond dimension O(Mk). In comparison, for just k=1, an exponentially larger bond dimension of 2M/6 is generically required for an MPS of qubits. We provide a simple QGN algorithm for approximate simulations of quantum dynamics in any spatial dimension. The approximate dynamics can achieve exact energy conservation for time-independent Hamiltonians, and spatial symmetries can also be maintained exactly. We benchmark the algorithm by simulating the quantum quench of fermionic Hamiltonians in up to three spatial dimensions.

Novel extended Halpern-type convergence algorithms for the split common fixed point problem involving α-demicontractive operators

As is well known, it is quite valuable and challenging to solve the split common fixed point problem (SCFP) associated with two different nonlinear operators. In this study, in order to solve (SCFP) with a more general setting of α-demicontractive operators due to Măruşter and Măruşter in 2011, we develop a kind of novel iterative approach in infinite dimensional Hilbert spaces. Further, using Xu's inequality and Maingé's lemma, strong convergence theorems of the suggested algorithms are established under some mild conditions. Finally, some special cases are considered, and the main results of this work are supported via two numerical comparisons.

Entangled coherent states and violations of Bell-CHSH inequalities

Three classes of entangled coherent states are employed to study the Bell-CHSH inequality. By using pseudospin operators in infinite dimensional Hilbert spaces, four dichotomic operators (A,A′,B,B′) entering the inequality are constructed. For each class of coherent states, we compute the correlator 〈ψ|AB+A′B+AB′−A′B′|ψ〉, analyzing the set of parameters that leads to a Bell-CHSH inequality violation and, particularly, to the saturation of Tsirelson's bound.

Bounded below composition operators on the space of Bloch functions on the unit ball of a Hilbert space

Let B_E be the open unit ball of a complex finite or infinite dimensional Hilbert space E and consider the space mathcal {B}(B_E) of Bloch functions on B_E. Using Lipschitz continuity of the dilation map on B_E given by x mapsto (1-Vert xVert ^2) mathcal {R}f(x) for x in B_E, where mathcal {R}f denotes the radial derivative of f in mathcal {B}(B_E), we study when a composition operator on mathcal {B}(B_E) is bounded below.

Unbounded Hamiltonians generated by Parseval frames

In Bagarello and Kużel (Acta Appl Math 171:4, 2021) Parseval frames were used to define bounded Hamiltonians, both in finite and in infinite dimesional Hilbert spaces. Here we continue this analysis, with a particular focus on the discrete spectrum of Hamiltonian operators constructed as a weighted infinite sum of rank one operators defined by some Parseval frame living in an infinite dimensional Hilbert space. The main difference with Bagarello and Kużel (Acta Appl Math 171:4, 2021) is that, here, the operators we consider are mostly unbounded. This is an useful upgrade with respect to our previous results, since physically meaningful Hamiltonians are indeed often unbounded. However, due to the fact that frames (in general) are not bases, the definition of an Hamiltonian is not so easy, and part of our results goes in this direction. Also, we analyze the eigenvalues of the Hamiltonians, and we discuss some physical applications of our framework.

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

We prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5–7], Fang and J. Xia [3] and the authors [8,9] on an open question posed by Pearcy in the seventies.

Bell Inequalities with Overlapping Measurements.

Which nonlocal correlations can be obtained, when a party has access to more than one subsystem? While traditionally nonlocality deals with spacelike separated parties, this question becomes important with quantum technologies that connect devices by means of small shared systems. Here, we study Bell inequalities where measurements of different parties can have overlap. This allows us to accommodate problems in quantum information such as the existence of quantum error correction codes in the framework of nonlocality. The scenarios considered show an interesting behavior with respect to Hilbert space dimension, overlap, and symmetry.

Quantum error correction in the black hole interior

We study the quantum error correction properties of the black hole interior in a toy model for an evaporating black hole: Jackiw-Teitelboim gravity entangled with a non-gravitational bath. After the Page time, the black hole interior degrees of freedom in this system are encoded in the bath Hilbert space. We use the gravitational path integral to show that the interior density matrix is correctable against the action of quantum operations on the bath which (i) do not have prior access to details of the black hole microstates, and (ii) do not have a large, negative coherent information with respect to the maximally mixed state on the bath, with the lower bound controlled by the black hole entropy and code subspace dimension. Thus, the encoding of the black hole interior in the radiation is robust against generic, low-rank quantum operations. For erasure errors, gravity comes within an O(1) distance of saturating the Singleton bound on the tolerance of error correcting codes. For typical errors in the bath to corrupt the interior, they must have a rank that is a large multiple of the bath Hilbert space dimension, with the precise coefficient set by the black hole entropy and code subspace dimension.

Quantum computing Floquet energy spectra

Quantum systems can be dynamically controlled using time-periodic external fields, leading to the concept of Floquet engineering, with promising technological applications. Computing Floquet energy spectra is harder than only computing ground state properties or single time-dependent trajectories, and scales exponentially with the Hilbert space dimension. Especially for strongly correlated systems in the low frequency limit, classical approaches based on truncation break down. Here, we present two quantum algorithms to determine effective Floquet modes and energy spectra. We combine the defining properties of Floquet modes in time and frequency domains with the expressiveness of parametrized quantum circuits to overcome the limitations of classical approaches. We benchmark our algorithms and provide an analysis of the key properties relevant for near-term quantum hardware.

Passive Quantum Measurement: Arrival Time, Quantum Zeno Effect and Gambler's Fallacy

AbstractClassical measurements are passive, in the sense that they do not affect the physical properties of the measured system. Normally, quantum measurements are not passive in that sense. In the infinite dimensional Hilbert space, however, it is found that quantum projective measurement can be passive in a way which is impossible in finite dimensional Hilbert spaces. Specifically, it is found that expectation value of a hermitian Hamiltonian can have an imaginary part in the infinite dimensional Hilbert space and that such an imaginary part implies a possibility to avoid quantum Zeno effect, which can physically be realized in quantum arrival experiments. The avoidance of quantum Zeno effect can also be understood as avoidance of a quantum version of gambler's fallacy, leading to the notion of passive quantum measurement that updates information about the physical system without affecting its physical properties. The arrival time probability distribution of a particle is found to be given by the flux of the probability current. Possible negative fluxes correspond to regimes at which there is no arrival at all, physically understood as regimes at which the particle departs rather than arrives.

Fast hybrid iterative schemes for solving variational inclusion problems

Tseng's forward‐backward‐forward splitting method for finding zeros of the sum of Lipschitz continuous monotone and maximal monotone operators is known to converge weakly in infinite dimensional Hilbert spaces. The inertial and viscosity approximation techniques are the techniques widely used to accelerate iterative algorithms and obtain strong convergence, respectively. In this paper, we propose two fast, strongly convergent modifications of Tseng's method and present some consequences and applications of our results. Moreover, we illustrate the performance and computability of our algorithms with relevant numerical examples. The two hybrid techniques incorporate the inertial and viscosity techniques. Unlike the conventional inertial‐viscosity hybrid techniques in the literature, our new algorithms compute both the inertial extrapolation and viscosity approximation simultaneously at the first step of each iteration.