arXiv subjects

Yoshifumi Nakata

Publications and source records attributed to Yoshifumi Nakata.

33 records · Page 2Linked to original sources

arXiv2020

Generic Entanglement Entropy for Quantum States with Symmetry

When a quantum pure state is drawn uniformly at random from a Hilbert space, the state is typically highly entangled. This property of a random state is known as generic entanglement of quantum states and has been long investigated from many perspectives, ranging from the black hole science to quantum information science. In this paper, we address the question of how symmetry of quantum states changes the properties of generic entanglement. More specifically, we study bipartite entanglement entropy of a quantum state that is drawn uniformly at random from an invariant subspace of a given symmetry. We first extend the well-known concentration formula to the one applicable to any subspace and then show that 1. quantum states in the subspaces associated with an axial symmetry are still highly entangled, though it is less than that of the quantum states without symmetry, 2. quantum states associated with the permutation symmetry are significantly less entangled, and 3. quantum states with translation symmetry are as entangled as the generic one. We also numerically investigate the phase-transition behavior of the distribution of generic entanglement, which indicates that the phase transition seems to still exist even when random states have symmetry.

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arXiv2018

Fundamental energy cost for quantum measurement

Measurements and feedback are essential in the control of any device operating at the quantum scale and exploiting the features of quantum physics. As the number of quantum components grows, it becomes imperative to consider the energetic expense of such elementary operations. Here, we derive energy requirements for general quantum measurement, extending previous models and obtaining stronger bounds in relevant situations, and then study two important classes of measurements in detail. One is the projective measurement, where we obtain the exact cost rather than a lower bound, and the other is the so-called inefficient measurement, in which we explicitly show that energy extraction is possible. As applications, we derive the energy-precision trade-off in quantum Zeno stabilisation schemes and the exact energy expense for quantum error correction. Our results constitute fundamental energetic limitations against which to benchmark implementations of future quantum devices as they grow in complexity.

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arXiv2017

Decoupling with random diagonal unitaries

We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli-$Z$ and -$X$ bases. This strategy was recently shown to achieve an approximate unitary $2$-design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary $2$-designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.

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arXiv2017

Unitary $2$-designs from random $X$- and $Z$-diagonal unitaries

Unitary $2$-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols, and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary $2$-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-$Z$ basis and that in the Pauli-$X$ basis. We also provide a converse about the number of repetitions needed to achieve unitary $2$-designs. These results imply that the process after $\ell$ repetitions achieves a $Θ(d^{-\ell})$-approximate unitary $2$-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary $2$-designs. Although a more efficient implementation of unitary $2$-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary $2$-design after switching interactions only a few times.

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arXiv2017

Efficient unitary designs with nearly time-independent Hamiltonian dynamics

We provide new constructions of unitary $t$-designs for general $t$ on one qudit and $N$ qubits, and propose a design Hamiltonian, a random Hamiltonian of which dynamics always forms a unitary design after a threshold time, as a basic framework to investigate randomising time evolution in quantum many-body systems. The new constructions are based on recently proposed schemes of repeating random unitaires diagonal in mutually unbiased bases. We first show that, if a pair of the bases satisfies a certain condition, the process on one qudit approximately forms a unitary $t$-design after $O(t)$ repetitions. We then construct quantum circuits on $N$ qubits that achieve unitary $t$-designs for $t = o(N^{1/2})$ using $O(t N^2)$ gates, improving the previous result using $O(t^{10}N^2)$ gates in terms of $t$. Based on these results, we present a design Hamiltonian with periodically changing two-local spin-glass-type interactions, leading to fast and relatively natural realisations of unitary designs in complex many-body systems.

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arXiv2014

Thermal states of random quantum many-body systems

We study a distribution of thermal states given by random Hamiltonians with a local structure. We show that the ensemble of thermal states monotonically approaches the unitarily invariant ensemble with decreasing temperature if all particles interact according to a single random interaction and achieves a state $t$-design at temperature $O(1/\log(t))$. For the system where the random interactions are local, we show that the ensemble achieves a state $1$-design. We then provide numerical evidence indicating that the ensemble undergoes a phase transition at finite temperature.

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arXiv2014

Diagonal quantum circuits: their computational power and applications

Diagonal quantum circuits are quantum circuits comprising only diagonal gates in the computational basis. In spite of a classical feature of diagonal quantum circuits in the sense of commutativity of all gates, their computational power is highly likely to outperform classical one and they are exploited for applications in quantum informational tasks. We review computational power of diagonal quantum circuits and their applications. We focus on the computational power of instantaneous quantum polynomial-time (IQP) circuits, which are a special type of diagonal quantum circuits. We then review an approximate generation of random states as an application of diagonal quantum circuits, where random states are an ensemble of pure states uniformly distributed in a Hilbert space. We also present a thermalizing algorithm of classical Hamiltonians by using diagonal quantum circuits. These applications are feasible to be experimentally implemented by current technology due to a simple and robust structure of diagonal gates.

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arXiv2014

Generating a state $t$-design by diagonal quantum circuits

We investigate protocols for generating a state $t$-design by using a fixed separable initial state and a diagonal-unitary $t$-design in the computational basis, which is a $t$-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary $t$-design generates a $O(1/2^N)$-approximate state $t$-design, where $N$ is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary $t$-design. We also show that it is necessary and sufficient to use $O(\log_2 t)$-qubit gates with random phases to generate a diagonal-unitary $t$-design by diagonal quantum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubit controlled-phase-type gates with discrete-valued random phases. Finally, we analyze the number of gates for implementing a diagonal-unitary $t$-design by {\it non-diagonal} two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.

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arXiv2014

Diagonal-unitary 2-designs and their implementations by quantum circuits

We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitary $t$-designs and present two quantum circuits that implement diagonal-unitary $2$-designs with the computational basis in $N$-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary $2$-design after applying the gates on all pairs of qubits. The number of required gates is $N(N-1)/2$. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact $2$-design, but achieves an $ε$-approximate $2$-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most $O(N^2(N+\log1/ε))$. We also provide an application of the circuits, a protocol of generating an exact $2$-design of random states by combining the circuits with a simple classical procedure requiring $O(N)$ random classical bits.

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arXiv2013

A general framework for phase and interference

Phase plays a crucial role in many quantum effects including interference. Phase is normally defined in terms of complex numbers that appear when representing quantum states as complex vectors. Here we give an operational definition whereby phase is instead defined in terms of measurement statistics. Our definition is phrased in terms of the operational framework known as generalised probabilistic theories, or the convex framework. The definition makes it possible to ask whether other theories in this framework can also have phase. We apply our definition to investigate phase and interference in several example theories: classical probability theory, a version of Spekken's toy model, quantum theory, and box-world. We find that phase is ubiquitous; any non-classical theory can be said to have non-trivial phase dynamics.

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arXiv2012

Measurement-Based Quantum Computation on Symmetry Breaking Thermal States

We consider measurement-based quantum computation (MBQC) on thermal states of the interacting cluster Hamiltonian containing interactions between the cluster stabilizers that undergoes thermal phase transitions. We show that the long-range order of the symmetry breaking thermal states below a critical temperature drastically enhance the robustness of MBQC against thermal excitations. Specifically, we show the enhancement in two-dimensional cases and prove that MBQC is topologically protected below the critical temperature in three-dimensional cases. The interacting cluster Hamiltonian allows us to perform MBQC even at a temperature an order of magnitude higher than that of the free cluster Hamiltonian.

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arXiv2012

Phase-random states: ensembles of states with fixed amplitudes and uniformly distributed phases in a fixed basis

Motivated by studies of typical properties of quantum states in statistical mechanics, we introduce phase-random states, an ensemble of pure states with fixed amplitudes and uniformly distributed phases in a fixed basis. We first show that canonical states typically appear in subsystems of phase-random states. We then investigate the simulatability of phase-random states, which is directly related to that of time evolution in closed systems, by studying their entanglement properties. We find that starting from a separable state, time evolutions under Hamiltonians composed of only separable eigenstates generate extremely high entanglement and are difficult to simulate with matrix product states. We also show that random quantum circuits consisting of only two-qubit diagonal unitaries can generate an ensemble with the same average entanglement as phase-random states.

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arXiv2012

Entanglement and the Interplay between Staggered Fields and Couplings

We investigate how the interplay between a staggered magnetic field and staggered coupling strength affects both ground state and thermal entanglement. Upon analytically calculating thermodynamic quantities and the correlation functions for such a system, we consider both the global Meyer-Wallach measure of entanglement and the concurrence between pairs of spins. We discover two quantum phase transitions present in the model and show that the quantum phase transitions are reflected in the behaviour of the entanglement at zero temperature. We discover that increasing the alternating field and alternating coupling strength can actually increase the amount of entanglement present at both zero temperature and for thermal states of the system.

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arXiv2011

Simulating typical entanglement with many-body Hamiltonian dynamics

We study the time evolution of the amount of entanglement generated by one dimensional spin-1/2 Ising-type Hamiltonians composed of many-body interactions. We investigate sets of states randomly selected during the time evolution generated by several types of time-independent Hamiltonians by analyzing the distributions of the amount of entanglement of the sets. We compare such entanglement distributions with that of typical entanglement, entanglement of a set of states randomly selected from a Hilbert space with respect to the unitarily invariant measure. We show that the entanglement distribution obtained by a time-independent Hamiltonian can simulate the average and standard deviation of the typical entanglement, if the Hamiltonian contains suitable many-body interactions. We also show that the time required to achieve such a distribution is polynomial in the system size for certain types of Hamiltonians.

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arXiv2009

Thermal robustness of multipartite entanglement of the 1-D spin 1/2 XY model

We study the robustness of multipartite entanglement of the ground state of the one-dimensional spin 1/2 XY model with a transverse magnetic field in the presence of thermal excitations, by investigating a threshold temperature, below which the thermal state is guaranteed to be entangled. We obtain the threshold temperature based on the geometric measure of entanglement of the ground state. The threshold temperature reflects three characteristic lines in the phase diagram of the correlation function. Our approach reveals a region where multipartite entanglement at zero temperature is high but is thermally fragile, and another region where multipartite entanglement at zero temperature is low but is thermally robust.

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