Unitary transformations formulate the time evolution of quantum states. How to learn a unitary transformation efficiently is a fundamental problem in quantum machine learning. The most natural and leading strategy is to train a quantum machine learning model based on a quantum dataset. Although the presence of more training data results in better models, using too much data reduces the efficiency of training. In this work, we solve the problem on the minimum size of sufficient quantum datasets for learning a unitary transformation exactly, which reveals the power and limitation of quantum data. First, we prove that the minimum size of a dataset with pure states is $2^n$ for learning an $n$-qubit unitary transformation. To fully explore the capability of quantum data, we introduce a practical quantum dataset consisting of $n+1$ elementary tensor product states that are sufficient for exact training. The main idea is to simplify the structure utilizing decoupling, which leads to an exponential improvement in the size of the datasets with pure states. Furthermore, we show that the size of the quantum dataset with mixed states can be reduced to a constant, which yields an optimal quantum dataset for learning a unitary. We showcase the applications of our results in oracle compiling and Hamiltonian simulation. Notably, to accurately simulate a 3-qubit one-dimensional nearest-neighbor Heisenberg model, our circuit only uses $96$ elementary quantum gates, which is significantly less than $4080$ gates in the circuit constructed by the Trotter-Suzuki product formula.
Quantum algorithms are typically understood in terms of the evolution of a multi-qubit quantum system under a prescribed sequence of unitary transformations. The input to the algorithm prescribes some of the unitary transformations in the sequence with others remaining fixed. For oracle query algorithms, the input determines the oracle unitary transformation. Such algorithms can be regarded as devices for discriminating amongst a set of unitary transformations. The question arises: "Given a set of known oracle unitary transformations, to what extent is it possible to discriminate amongst them?…
The equivalence of multipartite quantum mixed states under local unitary transformations is studied. A criterion for the equivalence of non-degenerate mixed multipartite quantum states under local unitary transformations is presented.
Quantum computation is based on implementing selected unitary transformations which represent algorithms. A generalized optimal control theory is used to find the driving field that generates a prespecified unitary transformation. The approach is illustrated in the implementation of one and two qubits gates in model molecular systems.
We show that several Hamiltonians that are $\mathcal{PT}$ symmetric may be taken to Hermitian Hamiltonians via a non-unitary transformation and vice versa. We also show that for some specific Hamiltonians such non-unitary transformations may be associated, via a fractional-Wick rotation, to complex time.
In terms of the analysis of fixed point subgroup and tensor decomposability of certain matrices, we study the equivalence of of quantum bipartite mixed states under local unitary transformations. For non-degenerate case an operational criterion for the equivalence of two such mixed bipartite states under local unitary transformations is presented.
We propose a genetic-algorithm-based method to find the unitary transformations for any desired quantum computation. We formulate a simple genetic algorithm by introducing the "genetic parameter vector" of the unitary transformations to be found. In the genetic algorithm process, all components of the genetic parameter vectors are supposed to evolve to the solution parameters of the unitary transformations. We apply our method to find the optimal unitary transformations and to generalize the corresponding quantum algorithms for a realistic problem, the one-bit oracle decision problem, or the o…
We concretely construct an extension of the controlled-U gate in qudit from some elementary gates. We also construct unitary transformation in two-qudit by means of the extended controlled-U gate and show the universality of it.
Quantum computation and quantum control operate by building unitary transformations out of sequences of elementary quantum logic operations or applications of control fields. This paper puts upper bounds on the minimum time required to implement a desired unitary transformation on a d-dimensional Hilbert space when applying sequences of Hamiltonian transformations. We show that strategies of building up a desired unitary out of non-infinitesimal and infinitesimal unitaries, or equivalently, using power and band limited controls, can yield the best possible scaling in time $O(d^2)$.
In the well known treatment of quantum teleportation, the receiver should convert the state of his EPR particle into the replica of the unknown quantum state by one of four possible unitary transformations. However, the importance of these unitary transformations must be emphasized. We will show in this paper that the receiver can not transform the state of his particle into an exact replica of the unknown state which the sender want to transfer if he have not a proper implementation of these unitary transformations. In the procedure of converting state, the inevitable coupling between EPR par…
We discuss the dynamical quantum systems which turn out to be bi-unitary with respect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbert space. We give a necessary and sufficient condition so that the Hermitian structures are in generic position. Finally the transformations of the bi-unitary group are explicitly obtained.
We investigate the equivalence of bipartite quantum mixed states under local unitary transformations by introducing representation classes from a geometrical approach. It is shown that two bipartite mixed states are equivalent under local unitary transformations if and only if they have the same representation class. Detailed examples are given on calculating representation classes.
A time-dependent unitary (canonical) transformation is found which maps the Hamiltonian for a harmonic oscillator with time-dependent real mass and real frequency to that of a generalized harmonic oscillator with time-dependent real mass and imaginary frequency. The latter may be reduced to an ordinary harmonic oscillator by means of another unitary (canonical) transformation. A simple analysis of the resulting system leads to the identification of a previously unknown class of exactly solvable time-dependent oscillators. Furthermore, it is shown how one can apply these results to establish a …
The aim of the article is to show how a coordinate transformation can be applied to the path-integral formalism. For this purpose the unitary definition of the quantum measure, which guarantees the conservation of total probability, is offered. As the examples, the phase space transformation to the canonically conjugate pare $(energy, time)$ and the transformation to the cylindrical coordinates are shown. The transformations of the path-integral measure looks classically but they can not be deduced from naive transformations of quantum trajectories.
We show that in principle, $N$-partite unitary transformations can be perfectly discriminated under local measurement and classical communication (LOCC) despite of their nonlocal properties. Based on this result, some related topics, including the construction of the appropriate quantum circuit together with the extension to general completely positive trace preserving operations, are discussed.
A quantum gate is realized by specific unitary transformations operating on states representing qubits. Considering a quantum system employed as an element in a quantum computing scheme, the task is therefore to enforce the pre-specified unitary transformation. This task is carried out by an external time dependent field. Optimal control theory has been suggested as a method to compute the external field which alters the evolution of the system such that it performs the desire unitary transformation. This study compares two recent implementations of optimal control theory to find the field tha…
In this paper, we study time-optimal control problems related to system of two coupled qubits where the time scales involved in performing unitary transformations on each qubit are significantly different. In particular, we address the case where unitary transformations produced by evolutions of the coupling take much longer time as compared to the time required to produce unitary transformations on the first qubit but much shorter time as compared to the time to produce unitary transformations on the second qubit. We present a canonical decomposition of SU(4) in terms of the subgroup SU(2)xSU…
The equivalence of tripartite pure states under local unitary transformations is investigated. The nonlocal properties for a class of tripartite quantum states in $\Cb^K \otimes \Cb^M \otimes \Cb^N$ composite systems are investigated and a complete set of invariants under local unitary transformations for these states is presented. It is shown that two of these states are locally equivalent if and only if all these invariants have the same values.
Final state of a black hole is studied by taking into account a general unitary evolution of a black hole matter state. It is found that the mixedness decreases under general unitary transformation when the initial matter state was in mixed state and the mean fidelity at the evaporation is smaller than the fidelity of the quantum teleportation by the factor of inverse square of the number of states of a black hole. The change of entanglement Alice-Bob pair at evaporation is also studied by calculating the entanglement fidelity and eigenvalues of the partial transposed block density matrix. We …
In this paper the geometry of two-qubit systems under local unitary group $SO(2)\otimes SU(2)$ is discussed. It is shown that the quaternionic conformal map intertwines between this local unitary subgroup of $Sp(2)$ and the quaternionic Möbius transformation which is rather a generalization of the results of Lee et al (2002 Quantum Inf. Process. 1 129).
We show that all pure entangled states of two $d$-dimensional quantum systems (i.e., two qudits) can be generated from an initial separable state via a universal Yang--Baxter matrix if one is assisted by local unitary transformations.
We proposed a general transformation in probabilistic teleportation, which is based on different entanglement matching coefficients $K$ corresponding to different unitary evolution which provides one with more flexible evolution method experimentally. Through analysis based on the Bell basis and generalized Bell basis measurement for two probabilistic teleportation, we suggested a general probability of successful teleportation, which is not only determined by the entanglement degree of transmission channels and measurement methods, but also related to the unitary transformation in the telepor…
After proving a general no-cloning theorem for black boxes, we derive the optimal universal cloning of unitary transformations, from one to two copies. The optimal cloner is realized by quantum channels with memory, and greately outperforms the optimal measure-and-reprepare cloning strategy. Applications are outlined, including two-way quantum cryptographic protocols.
We introduce a notion of Kolmogorov complexity of unitary transformation, which can (roughly) be understood as the least possible amount of information required to fully describe and reconstruct a given finite unitary transformation. In the context of quantum computing, it corresponds to the least possible amount of data to define and describe a quantum circuit or quantum computer program. Our Kolmogorov complexity of unitary transformation is built upon Kolmogorov "qubit complexity" of Berthiaume, W. Van Dam and S. Laplante via mapping from unitary transformations to positive operators, which…
We present a way of treating the problem of the interaction of a single trapped ion with laser beams based on successive aplications of unitary transformations onto the Hamiltonian. This allows the diagonalization of the Hamiltonian, by means of recursive relations, without performing the Lamb-Dicke approximation.
This paper reveals the relation between the canonical coset decomposition of unitary matrices and the corresponding decomposition via Householder reflections. These results can be used to parametrize unitary matrices via Householder reflections.
The equivalence of arbitrary dimensional bipartite states under local unitary transformations (LUT) is studied. A set of invariants and ancillary invariants under LUT is presented. We show that two states are equivalent under LUT if and only if they have the same values for all of these invariants.
In the present time we observe a growing number of publications where the, so-called, flow equations are successfully used to diagonalize Hamiltonians by means of an appropriate unitary transformation. Here we discuss and compare the flow equations (FE) method (proposed in 1994) with the method of one step continuous unitary transformations (OS CUT) (proposed in 1982). It is shown that the FE method can be considered as a generalization of the OS CUT approach to the case of parameter dependent generator. The OS CUT method gives linear differential equations for the diagonalization procedure. I…
Any lossless transformation on $n_{s}$ spatial and $n_{p}$ internal modes of light can be described by an $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix, but there is no known procedure to effect an arbitrary $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix on light in $n_{s}$ spatial and $n_{p}$ internal modes. We devise an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. Our realization uses beamsplitters and operations on internal modes to effect arbitrary linear transformations. The number of beamsplitters require…
It is generally accepted that no `faster than light signalling' (FTLS) using two entangled spin 1/2 particles is possible because of indeterminism in a quantum measurement and linearity of standard quantum mechanics. We show how in principle one bit of information could be transmitted using local measurements and a global unitary transformation of the state of two entangled spatially separated spin 1/2 particles. Assuming that the postulate of a state collapse due to measurement is valid, the no FTLS condition is saved if we do not have physical access to the required global unitary transforma…
We address the problem of finding the optimal joint unitary transformation on system + ancilla which is the most efficient in programming any desired channel on the system by changing the state of the ancilla. We present a solution to the problem for dim(H)=2 for both system and ancilla.
There are major advantages in a newer version of Grover's quantum algorithm utilizing a general unitary transformation in the search of a single object in a large unsorted database. In this paper, we generalize this algorithm to multiobject search. We show the techniques to achieve the reduction of the problem to one on an invariant subspace of dimension just equal to two.
In this paper we analyze a recent application of perturbation theory by the moment method to a family of two-dimensional anharmonic oscillators. By means of straightforward unitary transformations we show that two of the models studied by the authors are separable. Other is unbounded from below and therefore cannot be successfully treated by perturbation theory unless a complex harmonic frequency is introduced in the renormalization process. We calculate the lowest resonance by means of complex-coordinate rotation and compare its real part with the eigenvalue estimated by the authors. A pair o…
We consider the remote state creation via the homogeneous spin-1/2 chain and show that the significant extension of the creatable region can be achieved using the local unitary transformation of the so-called extended receiver (i.e. receiver joined with the nearest node(s)). This transformation is { most} effective in the models with all-node interactions. We consider the model with the two-qubit sender, one-qubit receiver and two-qubit extended receiver.
We present a novel approach for studying the dynamics of a superconducting qubit in a cavity. We succeed in linearizing the Hamiltonian through the application of an appropriate unitary transformation followed by a rotating wave approximation (RWA). For certain values of the parameters involved, we show that it is possible to obtain a a Jaynes-Cummings type Hamiltonian. As an example, we show the existence of super-revivals for the qubit inversion.
We address the problem of the information-disturbance trade-off associated to the estimation of a quantum transformation, and show how the extraction of information about the a black box causes a perturbation of the corresponding input-output evolution. In the case of a black box performing a unitary transformation, randomly distributed according to the invariant measure, we give a complete solution of the problem, deriving the optimal trade-off curve and presenting an explicit construction of the optimal quantum network.
A new form of local unitary (LU) transformation invariant is given for multi-qubit states . The general relation between tangle and the LU transformation invariant of pure three and four-qubit states is given. We find that the tangle actually is a special LU transformation invariant. Furthermore, the tangle is a important local unitary (LU) transformation invariant which is a nessesary condition for maximally multi-qubit entangled state.
Robust control design for quantum unitary transformations has been recognized as a fundamental and challenging task in the development of quantum information processing due to unavoidable decoherence or operational errors in the experimental implementation of quantum operations. In this paper, we extend the systematic methodology of sampling-based learning control (SLC) approach with a gradient flow algorithm for the design of robust quantum unitary transformations. The SLC approach first uses a "training" process to find an optimal control strategy robust against certain ranges of uncertainti…
The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms; but (ii) the required time for convergence to optimal con…
Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern …
The equivalence problem under local unitary transformation for $n$--partite pure states is reduced to the one for $(n-1)$--partite mixed states. In particular, a tripartite system $\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C$, where $\mathcal{H}_j$ is a finite dimensional complex Hilbert space for $j=A,B,C$, is considered and a set of invariants under local transformations is introduced, which is complete for the set of states whose partial trace with respect to $\mathcal{H}_A$ belongs to the class of generic mixed states.
In previous work the authors introduced a notion of generic states and obtained criteria for local equivalence of them. Here they introduce the concept of CHG states maintaining the criteria of local equivalence. This fact allows the authors to halve the number of invariants necessary to characterize the equivalence classes under local unitary transformations for the set of tripartite states whose partial trace with respect to one of the subsystems belongs to the class of CHG mixed states.
We give a one-to-one correspondence between classes of density matrices under local unitary invariance and the double cosets of unitary groups. We show that the interrelationship among classes of local unitary equivalent multi-partite mixed states is independent from the actual values of the eigenvalues and only depends on the multiplicities of the eigenvalues. The interpretation in terms of homogeneous spaces of unitary groups is also discussed.
An approach to the simulation of locally interacting systems is demonstrated and assayed. The proposal is built upon the concept of folding of bosonic modes previously introduced in the context of linear dynamics and can be seen as an alternative to Trotter-Susuki expansion in studies of quantum propagation based on tensor states. It is shown that evolution as well as ground state computations can be implemented and that test simulations deliver comparatively accurate results. The whole analysis provides insight into the way well-known quantum precursors affect mean values and fluctuations in …
We present a local optimal control strategy to produce desired unitary transformations. Unitary transformations are central to all quantum computational algorithms. Many realizations of quantum computation use a submanifold of states, comprising the quantum register, coupled by an external driving field to a collection of additional mediating excited states. Previous attempts to apply control theory to induce unitary transformations on the quantum register, while successful, produced pulses that drive the population out of the computational register at intermediate times. Leakage of population…
We construct a Hamiltonian for the generation of arbitrary pure states of the quantized electromagnetic field. The proposition is based upon the fact that a unitary transformation for the generation of number states has been already found. The general unitary transformation here obtained, would allow the use of nonlinear interactions for the production of pure states. We discuss the applicability of this method by giving examples of generation of simple superposition states. We also compare our Hamiltonian with the one resulting from the interaction of trapped ions with two laser fields.
A conditioned unitary transformation ($90^o$ polarization rotation) is performed at single-photon level. The transformation is realized by rotating polarization for one of the photons of a polarization-entangled biphoton state (signal photon) by means of a Pockel's cell triggered by the detection of the other (idler) photon after polarization selection. As a result, polarization degree for the signal beam changes from zero to the value given by the idler detector quantum efficiency. This result is relevant to practical realization of various quantum information schemes and can be used for deve…
The evolution of an open system is usually associated with the interaction of the system with an environment. A new method to study the open-type system evolution of a qubit (two-level atom) state is established. This evolution is determined by a unitary transformation applied to the qutrit (three-level atom) state, which defines the qubit subsystems. This procedure can be used to obtain different qubit quantum channels employing unitary transformations into the qutrit system. In particular, we study the phase damping and spontaneous-emission quantum channels. In addition, we mention a proposa…
The one-dimensional transverse field Ising model is solved by continuous unitary transformations in the high-field limit. A high accuracy is reached due to the closure of the relevant algebra of operators which we call string operators. The closure is related to the possibility to map the model by Jordan-Wigner transformation to non-interacting fermions. But it is proven without referring to this mapping. The effective model derived by the continuous unitary transformations is used to compute the contributions of one, two, and three elementary excitations to the diagonal dynamic structure fact…
Quantum Rabi model has been exactly solved by employing the parameter-dependent unitary transformation method in both the occupation number representation and the Bargmann space. The analytical expressions for the complete energy spectrum consisting of two double-fold degenerate sub-energy spectra are presented in the whole range of all the physical parameters. Each energy level is determined by a parameter in the unitary transformation, which obeys a highly nonlinear equation. The corresponding eigenfunction is a convergent infinite series in terms of the physical parameters. Due to the level…
Using the tool of unitary transformations of the extended receiver we perform simple operations with the non-diagonal elements of the initial sender's density matrix after their transferring to the receiver. These operations are following: turning some matrix elements to zero, rearranging the matrix elements and preparing their linear combinations with required coefficients. We also propose a protocol for solving a system of linear algebraic equations with a system of two equations as an example. Such operations are numerically simulated on the basis of 42-node spin-1/2 chain with the two-qubi…
Hypergraph states form a family of multiparticle quantum states that generalizes cluster states and graph states. We study the action and graphical representation of nonlocal unitary transformations between hypergraph states. This leads to a generalization of local complementation and graphical rules for various gates, such as the CNOT gate and the Toffoli gate. As an application, we show that already for five qubits local Pauli operations are not sufficient to check local equivalence of hypergraph states. Furthermore, we use our rules to construct entanglement witnesses for three-uniform hype…
The reliable and precise generation of quantum unitary transformations is essential to the realization of a number of fundamental objectives, such as quantum control and quantum information processing. Prior work has explored the optimal control problem of generating such unitary transformations as a surface optimization problem over the quantum control landscape, defined as a metric for realizing a desired unitary transformation as a function of the control variables. It was found that under the assumption of non-dissipative and controllable dynamics, the landscape topology is trap-free, impl…
