The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerant implementation of an $n$-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+T gates, incurring the T-count complexity of $O(n \log^2(n))$. In this paper, we show how to obtain approximate QFT with the T-count of $O(n \log(n))$. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.

Time has been an illusive concept to grasp. Although we do not yet understand it properly, there has been advances made in regards as to how we could explain it. One of such advances is the Page-Wootters' mechanism. In the mechanism time is seen as an inaccessible coordinate and the apparently passage of time arises as a consequence of correlations between the subsystems of a global state. Here we propose a measure that captures the relational character of the mechanism, showing that it is the internal coherence the necessary ingredient to the emergence of time in the Page-Wootters' model. Also, we connect it to results in quantum thermodynamics, showing that it is directly related to the extractable work from quantum coherence.

Several versions of quantum theory assume some form of localized collapse. If measurement outcomes are indeed defined by localized collapses, then a loophole-free demonstration of Bell non-locality needs to ensure space-like separated collapses associated with the measurements of the entangled systems. This collapse locality loophole remains largely untested, with one significant exception probing Diosi's and Penrose's gravitationally induced collapse hypotheses. I describe here techniques that allow much stronger experimental tests. These apply to all the well known types of collapse postulate, including gravitationally induced collapse, spontaneous localization models and Wigner's consciousness-induced collapse.

In the framework of Generalized probabilistic theories (GPT), we illustrate a class of statistical processes in case of two noninteracting identical particles in two modes that satisfies a well motivated notion of physicality conditions namely the double stochasticity and the no-interaction condition proposed by Karczewski et. al. (Phys. Rev. Lett. 120, 080401 (2018)), which can not be realized through a quantum mechanical process. This class of statistical process is ruled out by an additional requirement called the evolution condition imposed on two particle evolution. We also show that any statistical process of two noninteracting identical particles in two modes that satisfies all of the three physicality conditions can be realized within quantum mechanics using the beam splitter operation.

We study the dissipative preparation of pure non-Gaussian states of a target mode which is coupled both linearly and quadratically to an auxiliary damped mode. We show that any pure state achieved independently of the initial condition is either (i) a cubic phase state, namely a state given by the action of a non-Gaussian (cubic) unitary on a squeezed vacuum or (ii) a (squeezed and displaced) finite superposition of Fock states. Which of the two states is realized depends on whether the transformation induced by the engineered reservoir on the target mode is canonical (i) or not (ii). We discuss how to prepare these states in an optomechanical cavity driven with multiple control lasers, by tuning the relative strengths and phases of the drives. Relevant examples in (ii) include the stabilization of mechanical Schr\"odinger cat-like states or Fock-like states of any order. Our analysis is entirely analytical, it extends reservoir engineering to the non-Gaussian regime and enables the preparation of novel mechanical states with negative Wigner function.

The Hong-Ou-Mandel effect is considered a signature of the quantumness of light, as the dip in coincidence probability using semi-classical theories has an upper bound of 50%. Here we show, theoretically and experimentally, that, with proper phase control of the signals, classical pulses can mimic a Hong-Ou-Mandel-like dip. We demonstrate a dip of 99.635 +/- 0.002% with classical microwave fields. Quantumness manifests in wave-particle complementarity of the two-photon state. We construct quantum and classical interferometers for the complementarity test and show that while the two-photon state shows wave-particle complementarity, the classical pulses do not.

In this paper, we construct a new scheme for delegating a large circuit family, which we call "C+P circuits". "C+P" circuits are the circuits composed of Toffoli gates and diagonal gates. Our scheme is non-interactive, only requires small quantum resources on the client side, and can be proved secure in the quantum random oracle model, without relying on additional assumptions, for example, the existence of fully homomorphic encryption. In practice the random oracle can be replaced by appropriate hash functions or symmetric key encryption schemes, for example, SHA-3, AES.

This protocol allows a client to delegate the most expensive part of some quantum algorithms, for example, Shor's algorithm. The previous protocols that are powerful enough to delegate Shor's algorithm require either many rounds of interactions or the existence of FHE. The quantum resources required by the client are fewer than when it runs Shor's algorithm locally.

Different from many previous protocols, our scheme is not based on quantum one time pad, but on a new encoding called "entanglement encoding". We then generalize the garbled circuit to reversible garbled circuit to allow the computation on this encoding.

To prove the security of this protocol, we study key dependent message(KDM) security in the quantum random oracle model. Then as a natural generalization, we define and study quantum KDM security. KDM security was not previously studied in quantum settings.

Recent progress in photonics has led to a renewed interest in time-varying media that change on timescales comparable to the optical wave oscillation time. However, these studies typically overlook the role of material dispersion that will necessarily imply a delayed temporal response or, stated alternatively, a memory effect. We investigate the influence of the medium memory on a specific effect, i.e. the excitation of quantum vacuum radiation due to the temporal modulation. We construct a framework which reduces the problem to single-particle quantum mechanics, which we then use to study the quantum vacuum radiation. We find that the delayed temporal response changes the vacuum emission properties drastically: Frequencies mix, something typically associated with nonlinear processes, despite the system being completely linear. Indeed, this effect is related to the parametric resonances of the light-matter system, and to the parametric driving of the system by frequencies present locally in the drive but not in its spectrum.

Several experimental groups reported the evidence of multiple periodic modulations of nuclear decay constants which amplitudes are of the order 0.05% and have periods of one year, 24 hours or about one month. We argue that these deviations from radioactive decay law can be explained as the effect of small nonlinear corrections to standard quantum mechanics, in particular, to Hamiltonian of quantum system interaction with gravitational field. It's shown that modified Doebner-Goldin nonlinear model predicts the similar decay parameter variations under influence of Sun gravity.

A universal quantum computing scheme, with a universal set of logical gates, is proposed based on networks of 1D quantum systems. The encoding of information is in terms of universal features of gapped phases, for which effective field theories such as sine-Gordon field theory can be employed to describe a qubit. Primary logical gates are from twist, pump, glue, and shuffle operations that can be realized in principle by tuning parameters of the systems. Our scheme demonstrates the power of 1D quantum systems for robust quantum computing.

We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive energy photon. With invariant plane wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.

We propose a simple algorithm to convert a projected entangled pair state (PEPS) into a canonical form, analogous to the well-known canonical form of a matrix product state. Our approach is based on a variational gauging ansatz for the QR tensor decomposition of PEPS columns into a matrix product operator and a finite depth circuit of unitaries and isometries. We describe a practical initialization scheme that leads to rapid convergence in the QR optimization. We explore the performance and stability of the variational gauging algorithm in norm calculations for the transverse-field Ising and Heisenberg models on a square lattice. We also demonstrate energy optimization within the PEPS canonical form for the transverse-field Ising model. We expect this canonical form to open up improved analytical and numerical approaches for PEPS.

We prove a nearly tight lower bound on the approximate degree of the two-level $\mathsf{AND}$-$\mathsf{OR}$ tree using symmetrization arguments. Specifically, we show that $\widetilde{\mathrm{deg}}(\mathsf{AND}_m \circ \mathsf{OR}_n) = \widetilde{\Omega}(\sqrt{mn})$. To our knowledge, this is the first proof of this fact that relies on symmetrization exclusively; most other proofs involve formulating approximate degree as a linear program and exhibiting an explicit dual witness. Our proof relies on a symmetrization technique involving Laurent polynomials (polynomials with negative exponents) that was previously introduced by Aaronson, Kothari, Kretschmer, and Thaler [AKKT19].

Single photon detector (SPD) has a maximum count rate due to its dead time, which results in that the dynamic range of photon counting optical time-domain reflectometry (PC-OTDR) de-creases with the length of monitored fiber. To further improve the dynamic range of PC-OTDR, we propose and demonstrate an externally time-gated scheme. The externally time-gated scheme is realized by using a high-speed optical switch, i.e. a Mach-Zehnder interferometer, to modulate the back-propagation optical signal, and to allow that only a certain segment of the fiber is monitored by the SPD. The feasibility of proposed scheme is first examined with theoretical analysis and simulation; then we experimentally demonstrate it with our experimental PC-OTDR testbed operating at 800 nm wavelength band. In our studies, a dynamic range of 30.0 dB is achieved in a 70 meters long PC-OTDR system with 50 ns external gates, corresponding to an improvement of 11.0 dB in dynamic range comparing with no gating operation. Furthermore, with the improved dynamic range, a successful identification of a 0.37 dB loss event is detected with 30-seconds accumulation, which could not be identified without gating operation. Our scheme paves an avenue for developing PC-OTDR systems with high dynamic range.

We prove that the ground states of a local Hamiltonian satisfy an area law and can be computed in polynomial time when the interaction graph is a tree with discrete fractal dimension $\beta<2$. This condition is met for generic trees in the plane and for established models of hyperbranched polymers in 3D. This work is the first to prove an area law and exhibit a provably polynomial-time classical algorithm for local Hamiltonian ground states beyond the case of spin chains. Our algorithm outputs the ground state encoded as a multi-scale tensor network on the META-tree, which we introduce as an analogue of Vidal's MERA. Our results hold for polynomially degenerate and frustrated ground states, matching the state of the art for local Hamiltonians on a line.

We construct quantum MDS codes for quantum systems of dimension $q$ of length $q^2+1$ and minimum distance $d$ for all $d \leqslant q+1$, $d \neq q$. These codes are shown to exist by proving that there are classical generalised Reed-Solomon codes which are contained in their Hermitian-dual. These constructions include many constructions which were previously known but in some cases these codes appear to be new. We go on to prove that if $d\geqslant q+2$ then there in no generalised Reed-Solomon code which is contained in its Hermitian dual. We also construct a $ [\![ 18,0,10 ]\!]_5$ quantum MDS code, a $ [\![ 18,0,10 ]\!] _7$ quantum MDS code and a $ [\![ 14,0,8 ]\!]_5$ quantum MDS code, which are the first quantum MDS codes discovered for which $d \geqslant q+3$, apart from the $ [\![ 10,0,6 ]\!]_3$ quantum MDS code derived from Glynn's code.

Dynamical quantum phase transition (DQPT) is a periodic phase transition in large quantum systems wherein certain physical quantities show non-analyticity at a particular time $t_c$. We show by exact RG analysis of the quantum Ising model on scale invariant lattices of different dimensions that DQPT may involve fixed points of renormalization group which are unphysical in thermal phase transitions, and, in such cases, boundary conditions may become relevant. The transition points are determined exactly. A counter-example is also given that an equilibrium thermal phase transition does not necessarily imply a DQPT.

Vector vortex beams possess a topological property that derives both from the spatially varying amplitude of the field and also from its varying polarization. This property arises as a consequence of the inherent Skyrmionic nature of such beams and is quantified by the associated Skyrmion number. We illustrate this idea for some of the simplest vector beams and discuss the physical significance of the Skyrmion number in this context

Comprehending the dynamical behaviour of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood. Since the inception of Magnus' expansion in 1954, no fundamentally novel mathematical method for solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here of an entirely different non-perturbative approach, termed path-sum, which is always guaranteed to converge, yields the exact analytical solution in a finite number of steps for finite systems and is invariant under scale transformations of the quantum state space. Path-sum can be combined with any state-space reduction technique and can exactly reconstruct the dynamics of a many-body quantum system from the separate, isolated, evolutions of any chosen collection of its sub-systems. As examples of application, we solve analytically for the dynamics of all two-level systems as well as of a many-body Hamiltonian with a particular emphasis on NMR (Nuclear Magnetic Resonance) applications: Bloch-Siegert effect and $N$-spin systems involving the dipolar Hamiltonian and spin diffusion.

Quantum theory sets a bound on the minimal time evolution between initial and target states. This bound is called as quantum speed limit time. It is used to quantify maximal speed of quantum evolution. The quantum evolution will be faster, if quantum speed limit time decreases. In this work, we study the quantum speed limit time of a quantum state in the presence of disturbance effects in an environment. We use the model which is provided by Masashi Ban in \href{https://doi.org/10.1103/PhysRevA.99.012116}{Phys. Rev. A 99, 012116 (2019)}. In this model two quantum systems $\mathcal{A}$ and $\mathcal{S}$ interact with environment sequentially. At first, quantum system $\mathcal{A}$ interacts with the environment $\mathcal{E}$ as an auxiliary system then quantum system $\mathcal{S}$ interacts with disturbed environment immediately. In this work, we consider dephasing coupling with two types of environment with different spectral density: Ohmic and Lorentzian. We observe that, non-Markovian effects will be appear in the dynamics of quantum system $\mathcal{S}$ by the interaction of quantum system $\mathcal{A}$ with the environment. Given the fact that quantum speed limit time reduces due to non-Markovian effects, we show that disturbance effects will reduce the quantum speed limit time.