Unitary Gates Research Articles

Energy requirement for implementing unitary gates on energy-unbounded systems

The processing of quantum information always has a cost in terms of physical resources such as energy or time. Determining the resource requirements is not only an indispensable step in the design of practical devices—the resources need to be actually provided—but may also yield fundamental constraints on the class of processes that are physically possible. Here we study how much energy is required to implement a desired unitary gate on a quantum system with a non-trivial energy spectrum. We derive a general lower bound on the energy requirement, extending the main result of Chiribella et al (2021 Phys. Rev. X 11 021014) from finite dimensional systems to systems with unbounded Hamiltonians. Such an extension has immediate applications in quantum information processing with optical systems, and allows us to provide bounds on the energy requirement of continuous variable quantum gates, such as displacement and squeezing gates.

Classification of Schmidt-rank-two multipartite unitary gates by singular number

The multipartite unitary gates are called genuine if they are not product unitary operators across any bipartition. We mainly investigate the classification of genuine multipartite unitary gates of Schmidt rank two (SR-2), by focusing on the multiqubit scenario. For genuine multipartite (excluding bipartite) unitary gates of SR-2, there is an essential fact that their Schmidt decompositions are unique. Based on this fact, we propose a key notion named as singular number (SN) to classify the unitary gates concerned. The SN is defined as the number of local singular operators in the Schmidt decomposition. We then determine the accurate range of SN. For each SN, we formulate the parametric Schmidt decompositions of genuine multiqubit unitary gates under local equivalence. Finally, we extend the study to three-qubit diagonal unitary gates due to the close relation between diagonal unitary gates and SR-2 unitaries. We start with discussing two typical examples of SR-2, one of which is a fundamental three-qubit unitary gate, i.e. the CCZ gate. Then we characterize the diagonal unitary gates of SR greater than two. We show that a three-qubit diagonal unitary gate has SR at most three, and present a necessary and sufficient condition for such a unitary gate of SR-3. This completes the characterization of all genuine three-qubit diagonal unitary gates.

Complexity for superconformal primaries from BCH techniques

We extend existing results for the Nielsen complexity of scalar primaries and spinning primaries in four dimensions by including supersymmetry. Specifically, we study the Nielsen complexity of circuits that transform a superconformal primary with definite scaling dimension, spin and R-charge by means of continuous unitary gates from the mathfrak{su} (2, 2| mathcal{N} ) group. Our analysis makes profitable use of Baker-Campbell-Hausdorff formulas including a special class of BCH formulas we conjecture and motivate. With this approach we are able to determine the super-Kähler potential characterizing the circuit complexity geometry and obtain explicit expressions in the case of mathcal{N} = 1 and mathcal{N} = 2 supersymmetry.

Binary Optimal Control of Single-Flux-Quantum Pulse Sequences

We introduce a binary, relaxed gradient, trust-region method for optimizing pulse sequences for single flux quanta control of a quantum computer. The pulse sequences are optimized with the goal of realizing unitary gate transformations. Each pulse has a fixed amplitude and duration. We model this process as a binary optimal control problem, constrained by Schrödinger’s equation, where the binary variables indicate whether each pulse is on or off. We introduce a first-order trust-region method, which takes advantage of a relaxed gradient to determine an optimal pulse sequence that minimizes the gate infidelity, while also suppressing leakage to higher energy levels. The proposed algorithm has a computational complexity of , where is the number of pulses in the sequence. We present numerical results for the H and X gates, where the optimized pulse sequences give gate fidelities better than 99.9%, in trust-region iterations.

Transitions in the Learnability of Global Charges from Local Measurements.

We consider monitored quantum systems with a global conserved charge, and ask how efficiently an observer ("eavesdropper") can learn the global charge of such systems from local projective measurements. We find phase transitions as a function of the measurement rate, depending on how much information about the quantum dynamics the eavesdropper has access to. For random unitary circuits with U(1) symmetry, we present an optimal classical classifier to reconstruct the global charge from local measurement outcomes only. We demonstrate the existence of phase transitions in the performance of this classifier in the thermodynamic limit. We also study numerically improved classifiers by including some knowledge about the unitary gates pattern.

Maximal entanglement velocity implies dual unitarity

A global quantum quench can be modeled by a quantum circuit with local unitary gates. In general, entanglement grows linearly at a rate given by entanglement velocity, which is upper bounded by the growth of the light cone. We show that the unitary interactions achieving the maximal rate must remain unitary if we exchange the space and time directions -- a property known as dual unitarity. Our results are robust: approximate maximal entanglement velocity also implies approximate dual unitarity. We further show that maximal entanglement velocity is always accompanied by a specific dynamical pattern of entanglement, which yields simpler analyses of several known exactly solvable models.

Controlling the uncontrollable: Quantum control of open-system dynamics.

Control of open quantum systems is essential for the realization of contemporary quantum science and technology. We demonstrate such control using a thermodynamically consistent framework, taking into account the fact that the drive can modify the system's interaction with the environment. Such an effect is incorporated within the dynamical equation, leading to control-dependent dissipation. This relation serves as the key element for open-system control. The control paradigm is displayed by analyzing entropy-changing state-to-state transformations, such as heating and cooling. The difficult task of controlling quantum gates is achieved for nonunitary reset maps with complete memory loss. In addition, we identify a mechanism for controlling unitary gates by actively removing entropy from the system to the environment. We demonstrate a universal set of single- and double-qubit unitary gates under dissipation.

Linear-depth quantum circuits for multiqubit controlled gates

Quantum circuit depth minimization is critical for practical applications of circuit-based quantum computation. In this work, we present a systematic procedure to decompose multiqubit controlled unitary gates, which is essential in many quantum algorithms, to controlled-NOT and single-qubit gates with which the quantum circuit depth only increases linearly with the number of control qubits. Our algorithm does not require any ancillary qubits and achieves a quadratic reduction of the circuit depth against known methods. We show the advantage of our algorithm with proof-of-principle experiments on the IBM quantum cloud platform.

Noisy quantum computation modeled by quantum walk: universality without ancillas

The universal quantum-walk-based quantum computation model by Childs has opened the door for a new way of studying the limitations and advantages of quantum computation, as well as for its intermediate-term simulation. In recent years, the growing interest in noisy intermediate-scale quantum (NISQ) computers has lead to intense efforts being directed at understanding the computational advantages of open quantum systems. In this work, we extend the quantum walk model to open noisy systems in order to provide such a tool for the study of NISQ computers. Our method does not use explicit purification, and allows to ignore the environment degrees of freedom and obtain a much more efficient implementation (linear rather than exponential in the runtime), which employs no ancillas, hence provides direct access to the entanglement properties of the system. In our scheme, the quantum walk amplitudes represent elements of the density matrix rather than the wavefunction of a pure state. Despite the non-trivial manifestation of the normalization requirement in this setting, we model the application of general unitary gates and nonunitary channels, with an explicit implementation protocol for channels that are commonly used in noise models.

Measurement-based estimator scheme for continuous quantum error correction

Canonical discrete quantum error correction (DQEC) schemes use projective von Neumann measurements on stabilizers to discretize the error syndromes into a finite set, and fast unitary gates are applied to recover the corrupted information. Quantum error correction (QEC) based on continuous measurement, known as continuous quantum error correction (CQEC), in principle, can be executed faster than DQEC and can also be resource efficient. However, CQEC requires meticulous filtering of noisy continuous measurement data to reliably extract error syndromes on the basis of which errors could be detected. In this paper, we show that by constructing a measurement-based estimator (MBE) of the logical qubit to be protected, which is driven by the noisy continuous measurement currents of the stabilizers, it is possible to accurately track the errors occurring on the physical qubits in real time. We use this MBE to develop a continuous quantum error correction (MBE-CQEC) scheme that can protect the logical qubit to a high degree, surpassing the performance of DQEC, and also allows QEC to be conducted either immediately or in delayed time with instantaneous feedbacks.

Synthesis and Upper Bound of Schmidt Rank of Bipartite Controlled‐Unitary Gates

AbstractQuantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, it is shown generalized controlled‐X (GCX) gates, 6 single‐qubit rotations about the y‐ and z‐axes, and single‐partite y‐ and z‐rotation‐types which are defined in this paper are sufficient to simulate a controlled‐unitary gate with controlling on . In the scenario of the unitary gate with that is locally equivalent to a diagonal unitary on , GCX gates and single‐partite y‐ and z‐rotation‐types are required to simulate it. The quantum circuit for implementing and are presented. Furthermore, it is found that with controlling has Schmidt rank two, and in other cases the diagonalized form of the target unitaries can be expanded in terms of specific simple types of product unitary operators.

Construction and the ergodicity properties of dual unitary quantum circuits

We consider one-dimensional quantum circuits of the brickwork type, where the fundamental quantum gate is dual unitary. Such models are solvable: the dynamical correlation functions of the infinite temperature ensemble can be computed exactly. We review various existing constructions for dual unitary gates and we supplement them with new ideas in a number of cases. We discuss connections with various topics in physics and mathematics, including quantum information theory, tensor networks for the AdS/CFT correspondence (holographic error correcting codes), classical combinatorial designs (orthogonal Latin squares), planar algebras, and Yang-Baxter maps. Afterwards we consider the ergodicity properties of a special class of dual unitary models, where the local gate is a permutation matrix. We find an unexpected phenomenon: nonergodic behavior can manifest itself in multisite correlations, even in those cases when the one-site correlation functions are fully chaotic (completely thermalizing). We also discuss the circuits built out of perfect tensors. They appear locally as the most chaotic and most scrambling circuits, nevertheless they can show global signs of nonergodicity: if the perfect tensor is constructed from a linear map over finite fields, then the resulting circuit can show exact quantum revivals at unexpectedly short times. A brief mathematical treatment of the recurrence time in such models is presented in Appendix (written by Roland Bacher and Denis Serre).

Symmetric ternary quantum Fourier transform and its application

The research of ternary quantum system has gradually come into the attention of scholars in recent years. In $2018$, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In $2019$, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit flipping is realized by Not gate. Therefore, we need to extend Not gate to symmetric ternary system to realize qutrit flipping, which is called M-S gate. And then, by decomposing single-qutrit unitary gate in symmetric ternary quantum system, the universal gates are given. It means that any unitary operation on $n$ qutrits can be accurately implemented by single-qutrit symmetric ternary quantum gates and two-qutrit symmetric ternary M-S gates. By extending the QFT to the symmetric ternary quantum system, the paper successfully use some symmetric ternary quantum gates to construct the circuit which can realize symmetric ternary quantum Fourier transform (STQFT). Finally, the circuit of adder in symmetric ternary quantum system are designed based on the STQFT and the universal quantum gates. ternary quantum Fourier transform and its application (pp733-754) Hao Dong, Dayong Lu, and Xiaoyun Sun doi: https://doi.org/10.26421/QIC22.9-10-2 Abstracts: The research of ternary quantum system has gradually come into the attention of scholars in recent years. In $2018$, Guangcan Guo and his colleagues showed that in a qutrit-qutrit system they can observe quantum nonlocality and quantum contextuality at the same time. In $2019$, international cooperation team led by Anton Zelinger of Austrian Academy of Sciences and Jianwei Pan of University of science and technology of China, they have succeeded in teleporting complex high-dimensional quantum states. The work of the above scholars makes us clearly realize the importance of the study of ternary quantum system, but there is a few research results on this aspect. Furthermore, the quantum Fourier transform (QFT) offers an interesting way to perform arithmetic operations on a quantum computer. So, the paper extends the QFT to symmetric ternary quantum system and gives its applications. First, a set of quantum gates is defined for symmetric ternary quantum system. It is worth noting that in binary system, qubit flipping is realized by Not gate. Therefore, we need to extend Not gate to symmetric ternary system to realize qutrit flipping, which is called M-S gate. And then, by decomposing single-qutrit unitary gate in symmetric ternary quantum system, the universal gates are given. It means that any unitary operation on $n$ qutrits can be accurately implemented by single-qutrit symmetric ternary quantum gates and two-qutrit symmetric ternary M-S gates. By extending the QFT to the symmetric ternary quantum system, the paper successfully use some symmetric ternary quantum gates to construct the circuit which can realize symmetric ternary quantum Fourier transform (STQFT). Finally, the circuit of adder in symmetric ternary quantum system are designed based on the STQFT and the universal quantum gates.

Three-fold way of entanglement dynamics in monitored quantum circuits

We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson’s three circular ensembles (circular unitary, orthogonal, and symplectic ensembles; CUE, COE and CSE). We utilise the established model of a one-dimensional circuit evolving under alternating local random unitary gates and projective measurements performed with tunable rate, which for gates drawn from the CUE is known to display a transition from extensive to intensive entanglement scaling as the measurement rate is increased. By contrasting this case to the COE and CSE, we obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements. For this, we combine exact analytical random-matrix results for the entanglement generated by the individual gates in the different ensembles, and numerical results for the complete quantum circuit. These considerations include an efficient rephrasing of the statistical entangling power in terms of a characteristic entanglement matrix capturing the essence of Cartan’s KAK decomposition, and a general result for the eigenvalue statistics of antisymmetric matrices associated with the CSE.

Measurement-induced quantum phases realized in a trapped-ion quantum computer

Many-body open quantum systems balance internal dynamics against decoherence from interactions with an environment. Here, we explore this balance via random quantum circuits implemented on a trapped ion quantum computer, where the system evolution is represented by unitary gates with interspersed projective measurements. As the measurement rate is varied, a purification phase transition is predicted to emerge at a critical point akin to a fault-tolerent threshold. We probe the "pure" phase, where the system is rapidly projected to a deterministic state conditioned on the measurement outcomes, and the "mixed" or "coding" phase, where the initial state becomes partially encoded into a quantum error correcting codespace. We find convincing evidence of the two phases and show numerically that, with modest system scaling, critical properties of the transition clearly emerge.

Diagonal unitary and orthogonal symmetries in quantum theory: II. Evolution operators

We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.

Optimal universal quantum error correction via bounded reference frames

Error correcting codes with a universal set of transversal gates are a desideratum for quantum computing. Such codes, however, are ruled out by the Eastin-Knill theorem. Moreover, the theorem also rules out codes which are covariant with respect to the action of transversal unitary operations forming continuous symmetries. In this work, starting from an arbitrary code, we construct approximate codes which are covariant with respect to the entire group of local unitary gates in dimension $d\phantom{\rule{4pt}{0ex}}(<\ensuremath{\infty})$, using quantum reference frames. We show that our codes are capable of efficiently correcting different types of erasure errors. When only a small fraction of the $n$ qudits upon which the code is built are erased, our covariant code has an error that scales as $1/{n}^{2}$, which is reminiscent of the Heisenberg limit of quantum metrology. When every qudit has a chance of being erased, our covariant code has an error that scales as $1/n$. We show that the error scaling is optimal in both cases. Our approach has implications for fault-tolerant quantum computing, reference frame error correction, and the AdS-CFT duality.

From quantum speed limits to energy-efficient quantum gates

While recent breakthroughs in quantum computing promise the nascence of the quantum information age, quantum states remain delicate to control. Moreover, the required energy budget for large scale quantum applications has only sparely been considered. Addressing either of these issues necessitates a careful study of the most energetically efficient implementation of elementary quantum operations. In the present analysis, we show that this optimal control problem can be solved within the powerful framework of quantum speed limits. To this end, we derive state-independent lower bounds on the energetic cost, from which we find the universally optimal implementation of unitary quantum gates, for both single and N-qubit operations.

Noise-Induced Entanglement Transition in One-Dimensional Random Quantum Circuits

A random quantum circuit is a minimally structured model to study entanglement dynamics of many-body quantum systems. We consider a one-dimensional quantum circuit with noisy Haar-random unitary gates using density matrix operator and tensor contraction methods. It is shown that the entanglement evolution of the random quantum circuits is properly characterized by the logarithmic entanglement negativity. By performing exact numerical calculations, we find that, as the physical error rate is decreased below a critical value p c ≈ 0.056, the logarithmic entanglement negativity changes from the area law to the volume law, giving rise to an entanglement transition. The critical exponent of the correlation length can be determined from the finite-size scaling analysis, revealing the universal dynamic property of the noisy intermediate-scale quantum devices.

Measurement-induced criticality in extended and long-range unitary circuits

We explore the dynamical phases of unitary Clifford circuits with variable-range interactions, coupled to a monitoring environment. We investigate two classes of models, distinguished by the action of the unitary gates, which either are organized in clusters of finite-range two-body gates, or are pair-wise interactions randomly distributed throughout the system with a power-law distribution. We find the range of the interactions plays a key role in characterizing both phases and their measurement-induced transitions. For the cluster unitary gates we find a transition between a phase with volume-law scaling of the entanglement entropy and a phase with area-law entanglement entropy. Our results indicate that the universality class of the phase transition is compatible to that of short range hybrid Clifford circuits. Oppositely, in the case of power-law distributed gates, we find the universality class of the phase transition changes continuously with the parameter controlling the range of interactions. In particular, for intermediate values of the control parameter, we find a non-conformal critical line which separates a phase with volume-law scaling of the entanglement entropy from one with sub-extensive scaling. Within this region, we find the entanglement entropy and the logarithmic negativity present a cross-over from a phase with algebraic growth of entanglement with system size, and an area-law phase.