Current noisy intermediate-scale quantum (NISQ) devices can only execute small circuits with shallow depth, as they are still constrained by the presence of noise: quantum gates have error rates and quantum states are fragile due to decoherence. Hence, it is of great importance to optimize the depth/gate-count when designing quantum circuits for specific tasks. Diagonal unitary matrices are well-known to be key building blocks of many quantum algorithms or quantum computing procedures. Prior work has discussed the synthesis of diagonal unitary matrices over the primitive gate set $\{\text{CNOT}, R_Z\}$. However, the problem has not yet been fully understood, since the existing synthesis methods have not optimized the circuit depth.
In this paper, we propose a depth-optimized synthesis algorithm that automatically produces a quantum circuit for any given diagonal unitary matrix. Specially, it not only ensures the asymptotically optimal gate-count, but also nearly halves the total circuit depth compared with the previous method. Technically, we discover a uniform circuit rewriting rule well-suited for reducing the circuit depth. The performance of our synthesis algorithm is both theoretically analyzed and experimentally validated by evaluations on two examples. First, we achieve a nearly 50\% depth reduction over Welch's method for synthesizing random diagonal unitary matrices with up to 16 qubits. Second, we achieve an average of 22.05\% depth reduction for resynthesizing the diagonal part of specific quantum approximate optimization algorithm (QAOA) circuits with up to 14 qubits.
We give a new quantum circuit approximation of quantum multiplexors based on the idea of complexity theory oracles. As an added bonus, our multiplexor approximation immediately gives a quantum circuit approximation of diagonal unitary matrices.
We classify, up to local unitary equivalence, the set of $n$-qubit states that is stabilized by the diagonal subgroup of the local unitary group. We exhibit a basis for this set, parameterized by diagrams of nonintersecting chords connecting pairs of points on a circle, and give a criterion for when the stabilizer is precisely the diagonal subgroup and not larger. This investigation is part of a larger program to partially classify entanglement type (local unitary equivalence class) via analysis of stabilizer structure.
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 di…
Off-diagonal mixed state phases based upon a concept of orthogonality adapted to unitary evolution and a proper normalisation condition are introduced. Some particular instances are analysed and parallel transport leading to the off-diagonal mixed state geometric phase is delineated. A complete experimental realisation of the off-diagonal mixed state geometric phases in the qubit case using polarisation-entangled two-photon interferometry is proposed.
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 …
We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in …
We investigate the geometric phases and the Bargmann invariants associated with a multi-level quantum systems. In particular, we show that a full set of `gauge-invariant' objects for an $n$-level system consists of $n$ geometric phases and ${1/2}(n-1)(n-2)$ algebraically independent 4-vertex Bargmann invariants. In the process of establishing this result we develop a canonical form for U(n) matrices which is useful in its own right. We show that the recently discovered `off-diagonal' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analysed in te…
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 …
Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University（尼古拉·哥白尼大学物理、天文与信息学院物理研究所）；Tata Institute of Fundamental Research Hyderabad（塔塔基础研究所海得拉巴分所）；Institute for Quantum Studies, Chapman University（查普曼大学量子研究学院）；Department of Computer Science, Texas Tech University（德克萨斯理工大学计算机科学系）
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. …
We introduce a generalization of the set of completely positive matrices that we call "pairwise completely positive" (PCP) matrices. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. We explore basic properties of these matrix pairs and develop several testable necessary and sufficient conditions that help determine whether or not a pair is PCP. We then establish a connection with quantum entanglement by showing that determining whether or not a pair of matrices is pai…
We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenv…
Any unitary transformation of quantum computational networks is explicitly decomposed, in an exact and unified form, into a sequence of a limited number of one-qubit quantum gates and the two-qubit diagonal gates that have diagonal unitary representation in usual computational basis. This decomposition may be simplified greatly with the help of the properties of the finite-dimensional multiple-quantum operator algebra spaces of a quantum system and the specific properties of a given quantum algorithm. As elementary building blocks of quantum computation, the two-qubit diagonal gates and one-qu…
Floquet theory combined with the generalized Van Vleck nearly degenerate perturbation theory, has been widely employed for studying various two-level systems that are driven by external fields via the time-dependent longitudinal (i.e., diagonal) couplings. However, three-level systems strongly driven by the time-dependent transverse (i.e., off-diagonal) couplings have rarely been investigated, due to the breakdown of the traditional rotating wave approximation. Meanwhile, the conventional perturbation theory is not directly applicable, since a small parameter for the perturbed part is no longe…
In this paper we discuss the utilization of Variational Quantum Solver (VQE) and recently introduced Generalized Unitary Coupled Cluster (GUCC) formalism for the diagonalization of downfolded/effective Hamiltonians in active spaces. In addition to effective Hamiltonians defined by the downfolding of a subset of virtual orbitals we also consider their form defined by freezing core orbitals, which enables us to deal with larger systems. We also consider various solvers to identify solutions of the GUCC equations. We use N$_2$, H$_2$O, and C$_2$H$_4$, and benchmark systems to illustrate the perfo…
Realizing non-unitary transformations on unitary-gate based quantum devices is critically important for simulating a variety of physical problems including open quantum systems and subnormalized quantum states. We present a dilation based algorithm to simulate non-unitary operations using probabilistic quantum computing with only one ancilla qubit. We utilize the singular-value decomposition (SVD) to decompose any general quantum operator into a product of two unitary operators and a diagonal non-unitary operator, which we show can be implemented by a diagonal unitary operator in a 1-qubit dil…
The unitary generation of coherence from an incoherent thermal state is investigated. We consider a completely controllable Hamiltonian allowing to generate all possible unitary transformations. Optimizing the unitary control to achieve maximum coherence leads to a micro-canonical energy distribution on the diagonal energy representation. We demonstrate such a control scenario starting from a Hamiltonian utilizing optimal control theory for unitary targets. Generating coherence from an incoherent initial state always costs external work. By constraining the amount of work invested by the contr…
In this paper, we work on a quantum walk whose system is manipulated by a five-diagonal unitary matrix, and present long-time limit distributions. The quantum walk launches off a location and delocalizes in distribution as its system is getting updated. The five-diagonal matrix contains a phase term and the quantum walk becomes a standard coined walk when the phase term is fixed at special values. Or, the phase term gives an effect on the quantum walk. As a result, we will see an explicit form of a long-time limit distribution for a quantum walk driven by the matrix, and thanks to the exact fo…
State preparation is a process encoding the classical data into the quantum systems. Based on quantum phase estimation, we propose the specific quantum circuits for a deterministic state preparation algorithm and a probabilistic state preparation algorithm. To discuss the gate complexity in these algorithms, we decompose the diagonal unitary operators included in the phase estimation algorithms into the basic gates. Thus, we associate the state preparation problem with the decomposition problem of the diagonal unitary operators. We analyse the fidelities in the two algorithms and discuss the s…
The accurate evaluation of diagonal unitary operators is often the most resource-intensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we point out a correspondence between Walsh functions and a basis for diagonal operators that gives a simple way to construct efficient circuits for diagonal unitaries without ancillas. This correspondence reduces the problem of constructing the minimal-depth circuit wi…
We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with an unitary irreducible representation of a (compact) Lie group. We show that necessary and sufficient conditions for the possibility of such a representation can be obtained by combining Clebsch-Gordan theory and the reciprocity theorems associated with induced unitary group representation. Applications to several examples involving $SU(2),$ $SU(3),$ and the Heisenberg-Weyl group are presented, showing that there are simple examples of genera…
Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $ρ=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study t…
We generalize the concepts of Internal Time Superoperator, its associated non unitary similarity transformations and Liapounov variables, to quantum systems with diagonal singularity, and we give a constructive proof of the existence of these superoperators for systems with purely diagonal Hamiltonian having uniform absolutely continuous spectrum on the interval from zero to infinity.
Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into two diagonal unitaries and one whose row- and column sums are equal to one. The proof is non-constructive and based on a reformulation in terms of symplectic topology. As a corollary, we obtain a decomposition of unitary matrices into an interlaced product of unitary diagonal matrices and discrete Fourier transformations. This provides a new decomposition of…
We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our …
We report progress on the LU-LC conjecture - an open problem in the context of entanglement in stabilizer states (or graph states). This conjecture states that every two stabilizer states which are related by a local unitary operation, must also be related by a local operation within the Clifford group. The contribution of this paper is a reduction of the LU-LC conjecture to a simpler problem - which, however, remains to date unsolved. As our main result, we show that, if the LU-LC conjecture could be proved for the restricted case of diagonal local unitary operations, then the conjecture is c…
Quantum correlations in the state of four-level atom are investigated by using generic unitary transforms of the classical (diagonal) density matrix. Partial cases of pure state, $X$-state, Werner state are studied in details. The geometrical meaning of unitary Hilbert reference-frame rotations generating entanglement in the initially separable state is discussed. Characteristics of the entanglement in terms of concurrence, entropy and negativity are obtained as functions of the unitary matrix rotating the reference frame.
Asymptotically Optimal Circuit Depth for Diagonal Unitary Synthesis and Compilation on Two-Dimensional Grids
School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；College of Information Engineering, China Jiliang University（中国计量大学信息工程学院）；Key Laboratory of Computer Network and Information Integration (Southeast University), Ministry of Education（教育部计算机网络与信息集成重点实验室（东南大学））；Nanjing Meteorology Bereau（南京市气象局）
We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular…
Quantum circuit model is the most popular paradigm for implementing complex quantum computation. Based on Cartan decomposition, we show that $2(N-1)$ generalized controlled-$X$ (GCX) gates, $6$ single-qubit rotations about the $y$- and $z$-axes, and $N+5$ single-partite $y$- and $z$-rotation-types which are defined in this paper are sufficient to simulate a controlled-unitary gate $\mathcal{U}_{cu(2\otimes N)}$ with $A$ controlling on $\mathbb{C}^2\otimes \mathbb{C}^N$. In the scenario of the unitary gate $\mathcal{U}_{cd(M\otimes N)}$ with $M\geq3$ that is locally equivalent to a diagonal uni…
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 …
The diagonal entropy was introduced as a good entropy candidate especially for isolated quantum systems out of equilibrium. Here we present an analytical calculation of the average diagonal entropy for systems undergoing unitary evolution and an external perturbation in the form of a cyclic quench. We compare our analytical findings with numerical simulations of various many-body quantum systems. Our calculations elucidate various heuristic relations proposed recently in the literature.
We study the structure of bipartite unitary operators which generate via the Stinespring dilation theorem, quantum operations preserving some given matrix algebra, independently of the ancilla state. We characterize completely the unitary operators preserving diagonal, block-diagonal, and tensor product algebras. Some unexpected connections with the theory of quantum Latin squares are explored, and we introduce and study a Sinkhorn-like algorithm used to randomly generate quantum Latin squares.
We study a deformation of the counterdiabatic-driving Hamiltonian as a systematic strategy for an adiabatic control of quantum states. Using a unitary transformation, we design a convenient form of the driver Hamiltonian. We apply the method to a particle in a confining potential and discrete systems to find explicit forms of the Hamiltonian and discuss the general properties. The method is derived by using the quantum brachistochrone equation, which shows the existence of a nontrivial dynamical invariant in the deformed system.
Vibrational Sample-Based Quantum Diagonalization with Unitary-Cluster-Jastrow Ansätze on IBM QPUs
IBM Quantum, IBM Japan（日本IBM 量子计算部门）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所 跨学科理论与数学科学中心（iTHEMS））；RIKEN（理化学研究所）；RIKEN Pioneering Research Institute (PRI)（理化学研究所 前沿研究机构（PRI））；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所 新兴物质科学中心（CEMS））；RIKEN Center for Computational Science (R-CCS)（理化学研究所 计算科学中心（R-CCS））；RIKEN Center for Quantum Computing (RQC)（理化学研究所 量子计算中心（RQC））
In this work we present a method of decomposition of arbitrary unitary matrix $U\in\mathbf U(2^k)$ into a product of single-qubit negator and controlled-$\sqrt{\mbox{NOT}}$ gates. Since the product results with negator matrix, which can be treated as complex analogue if bistochastic matrix, our method can be seen as complex analogue of Sinkhorn-Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively and resulting circuit consists of $O(4^k)$ entangling gates, which is proved to be optimal. An example of su…
Symplectic split-operator method for the time-dependent unitary Tavis-Cummings model
Akhiezer Institute for Theoretical Physics, NSC KIPT（阿基谢尔理论物理研究所，NSC KIPT）；Advanced Photonic and Electronic Sciences Division, U.S. Army DEVCOM Army Research Laboratory（先进光子与电子科学分部，美国陆军DEVCOM陆军研究实验室）；Department of Physics, University of Massachusetts at Boston（马萨诸塞大学波士顿分校物理系）；Education and Research Institute “School of Physics and Technology”, Karazin Kharkiv National University（克拉辛哈尔科夫国立大学“物理与技术学院”教育与研究所）；Department of Physics and Engineering Physics, Tulane University（杜兰大学物理与工程物理系）
While implementing a quantum algorithm it is crucial to reduce the quantum resources, in order to obtain the desired computational advantage. For most fault-tolerant quantum error-correcting codes the cost of implementing the non-Clifford gate is the highest among all the gates in a universal fault-tolerant gate set. In this paper we design provable algorithm to determine T-count of any $n$-qubit ($n\geq 1$) unitary $W$ of size $2^n\times 2^n$, over the Clifford+T gate set. The space and time complexity of our algorithm are $O\left(2^{2n}\right)$ and $O\left(2^{2n\mathcal{T}_ε(W)+4n}\right)$ r…
Effective low-energy theories represent powerful theoretical tools to reduce the complexity in modeling interacting quantum many-particle systems. However, common theoretical methods rely on perturbation theory, which limits their applicability to weak interactions. Here we introduce the Variational Adiabatic Gauge Transformation (VAGT), a non-perturbative hybrid quantum algorithm that can use nowadays quantum computers to learn the variational parameters of the unitary circuit that brings the Hamiltonian to either its block-diagonal or full-diagonal form. If a Hamiltonian can be diagonalized …
We construct $(2n)^2\times (2n)^2$ unitary braid matrices $\hat{R}$ for $n\geq 2$ generalizing the class known for $n=1$. A set of $(2n)\times (2n)$ matrices $(I,J,K,L)$ are defined. $\hat{R}$ is expressed in terms of their tensor products (such as $K\otimes J$), leading to a canonical formulation for all $n$. Complex projectors $P_{\pm}$ provide a basis for our real, unitary $\hat{R}$. Baxterization is obtained. Diagonalizations and block-diagonalizations are presented. The loss of braid property when $\hat{R}$ $(n>1)$ is block-diagonalized in terms of $\hat{R}$ $(n=1)$ is pointed out and exp…
We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of $i=\sqrt{-1}$ times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudo-unitary matrices. In particular, we present a thorough treatment of $2\times 2$ pseudo-unitary matrices and discuss an example of a quantum system with a $2\times 2$ …
We have proposed and demonstrated a scalable and efficient scheme for programmable unitary operations in orbital angular momentum (OAM) domain. Based on matrix decomposition into diagonal and Fourier factors, arbitrary matrix operators can be implemented only by diagonal matrices alternately acting on orbital angular momentum domain and azimuthal angle domain, which are linked by Fourier transform. With numerical simulations, unitary matrices with dimensionality of 3*3 are designed and discussed for OAM domain. Meanwhile, the parallelism of our proposed scheme is also presented with two 3*3 ma…
A microscopic definition of the thermodynamic entropy in an isolated quantum system must satisfy (i) additivity, (ii) extensivity and (iii) the second law of thermodynamics. We show that the diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, meets the first two requirements and that the third requirement is satisfied if an arbitrary external operation is performed at typical times. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that the Hamiltonian dynamics restricts quantum trajectories under unitary evoluti…
Motivated by recent results in multiphoton interferometry, we expand a result of Kostant on immanants of an arbitrary $m\times m$ unitary matrix $T\in$ su$(m)$ to the submatrices of $T$. Specifically, we show that immanants of principal submatrices of a unitary matrix $T$ are a sum $\sum_{t} D^{(λ)}_{tt}(Ω)$ of the diagonal $D$-functions of group element $Ω$, with $t$ determined {by} the choice of submatrix, and the irrep $(λ)$ determined by the immanant under consideration. We also provide evidence that this result extends to some submatrices that are not principal diagonal, and we discuss ho…
In quantum gravity the unitary evolution does not follow from the Wheeler-DeWitt dynamics equation as it follows from the Schrödinger equation in non-relativistic quantum mechanics. Therefore we can define a spin-foam model based on SL(2,C) spinor finite non-unitary representations. The recently discovered holomorphic Peter-Weyl theorem \cite{Huebschmann} made it possible to decompose the delta function of a non-compact Lorentz group into the convergent sum of the matrix coefficients. We calculate the vertex amplitude with the help of that theorem and obtain a simple expression for our model. …
We study the density of states (DOS) for disordered systems whose spectral statistics can be described by a Gaussian ensemble of almost diagonal Hermitian random matrices. The matrices have independent random entries $ H_{i \geq j} $ with small off-diagonal elements: $ <|H_{i \neq j}|^{2} > \ll <|H_{ii}|^{2} > \sim 1 $. Using the recently suggested method of a {\it virial expansion in the number of interacting energy levels} (Journ.Phys.A {\bf 36}, 8265), we calculate the leading correction to the Poissonian DOS in the cases of the Gaussian Orthogonal and Unitary Ensembles. We apply the genera…
We describe a compilation strategy for Variational Quantum Eigensolver (VQE) algorithms which use the Unitary Coupled Cluster (UCC) ansatz, designed to reduce circuit depth and gate count. This is achieved by partitioning Pauli exponential terms into mutually commuting sets. These sets are then diagonalised using Clifford circuits and synthesised using the phase polynomial formalism. This strategy reduces cx depth by 75.4% on average, and by up to 89.9%, compared to naive synthesis for a variety of molecules, qubit encodings and basis sets.
The orbit tracking of free-evolutionary target system in closed quantum systems is studied in this paper. Based on the concept of system control theory, the unitary transformation is applied to change the time-dependent target function into a stationary target state so that the orbit tracking problem is changed into the state transfer one. A Lyapunov function with virtual mechanical quantity P is employed to design a control law for such a state transferring. The target states in density matrix are grouped into two classes: diagonal and non-diagonal. The specific convergent conditions for targ…
