Engineering of arbitrary U(N) transformation by quantum Householder reflections P. A. Ivanov, E. S. Kyoseva, and N. V. Vitanov

Plan of the talk Standard, generalized and coupled Householder Reflections Physical Implementations of standard and generalized HRs Decomposition of U(N) group by standard and generalized HRs Navigation in Hilbert space by HRs Physical Implementation of coupled HR Decomposition of U(N) group by coupled HR

For an arbitrary vector we wish to find another vector, which is the reflection of on the other side of the plane P REFLECTION STANDARD HOUSEHOLDER REFLECTION A. S. Householder, J. ACM 5, 339 (1958)

PROPERTIES REFLECTION 1.If we reflect a vector twice through the same plane, we get the same vector again 2. M is equal to its inverse GENERALIZED HR The standard HR is a special case of the generalized HR for COUPLED HR

APPLICATION of HR Householder decomposition of matricesHouseholder decomposition of matrices Solving systems of linear equationsSolving systems of linear equations Finding eigenvalues of high-dimensional matricesFinding eigenvalues of high-dimensional matrices Least-square optimizationLeast-square optimization Grover algorithmGrover algorithm

Physical Implementations of the standard and generalized Householder Reflections We consider (N+1)-state system with N degenerate ground states, which represent the qunit, coupled coherently via a common excited state by pulsed external fields of the same time dependence but possible different amplitudes and phases. Such an N-pod system can be formed, e.g., by coupling the magnetic sublevels of several J=1 levels to a single J=0 level; for a qutrit only one J=1 level suffices Phys. Rev. A (2006) HAMILTONIAN OF THE SYSTEM Rabi frequency

Morris- Shore transformation The coupled (N+1)-state system can be decomposed into a set of N-1 dark ground states, and a two-state system, consisting of a bright state and the excited state Morris- Shore transformation Phys. Rev. A (1983)

Schrödinger Equation The exact solution for the propagator Cayley-Klein parameters

STANDARD QHR: Exact resonance Pulse Area The propagator reduces to: U is an N-dimensional unitary matrix, which represents the propagator within the N- state degenerate manifold, which has exactly the standard QHR form : The components of the vector are the normalized Rabi frequencies The propagator U of the N- pod system driven by Hamiltonian represents a compact physical realization of standard QHR in a single interaction step.

Generalized QHR: Rosen- Zener model The Rozen - Zener propagator Physical realization of the generalized QHR The Rozen- Zener model can be seen as an extension of the resonance solution to nonzero detuning for a special pulse shape (hyperbolic secant ) The phase depends on the detuning and for an arbitrary integer l we find The use of nonresonant interaction, besides an additional phase parameter, has another important advantage over resonant pulses: lower transient population of the intermediate state. This can be crucial if the lifetime of this state is short compared to the interaction duration

Engineering of arbitrary U(N) transformation by rotations The matrix a beam-splitter with a phase shifter Any N-dimensional unitary matrix can be represented as product N(N-1)/2 rotations and one phase transformation Phys. Lett (1994)

Householder Decomposition Standard QHR decomposition Any N-dimensional unitary matrix U can be expressed as a product of N-1 standard QHRs and one phase gate Generalized QHR decomposition Any N-dimensional unitary matrix U can be expressed as a product of N generalized QHRs Phys. Rev. A (2006)

Proof of the Standard QHR decomposition 1. First we define the normalized vector It is easy to see Hence the action of M upon U nullifies the first row and the first column except for the first element nth column of U

2. We repeat the same procedure on MU and construct the vector The corresponding QHR applied to MU, has the following effects: (i)it nullifies the second row and second column of MU except for the diagonal element, and (ii) does not change the first row and first column. After N-1 interaction steps By repeating the same procedure N-1 times, we construct N-1 consecutive Householder reflections, which nullify all off-diagonal elements, to produce a diagonal matrix comprising N phase factors

Proof of the Generalized QHR decomposition 1.We first define the normalized vector It is readily seen that Therefore, the action of the firs generalized HR upon U nullifies the first row and first column except for the first element, which is turned into unit

2.We repeat the same procedure The corresponding QHR applied to MU, has the following effects :(i) it nullifies the second row and second column of MU except for the diagonal element which is turned into unit, and (ii) does not change the first row and first column. By repeating the same procedure N times, we construct N consecutive Householder reflection, which nullify all off-diagonal elements, to produce the identity matrix

COMMENT 2. The QHR decomposition of the U(N) group provides a simple and efficient physical realization of a general transformation of a qunit by only N interaction steps; this is a significant advance compared to O(N 2 ) operations in existing recipes. 3. Each QHR vectors is N-dimensional, but the nonzero elements decrease 4. The decomposition is also of mathematical interest because it provide a very natural parametrization of the U(N) group. 1.The choice of the QHRs is not unique: for example, the first QHR vector can be constructed from first row, instead of the first column. Furthermore, the final diagonal matrix occurs due to the unitarity of U; QHR sequence produces a triangular matrix in general. from N in the first step to just 1 in the final step

Quantum Fourier Transform For Qubit, the QFT can be written as a single Householder reflection Physical Implementation EXAMPLES

For Qutrit, the QFT can be written as a product of two standard QHRs and one phase gate, or as a product of two generalized QHRs

NAVIGATION IN HILBERT SPACE Given the initial state and final state of the qunit, we wish to find propagator such that Phys. Rev. A (2007) Transition by standard QHRs 1. First we define the normalized vector The corresponding QHR acting upon the initial state reflects it onto single qunit state

NAVIGATION IN HILBERT SPACE 2. Next, we define the vector The action of corresponding standard QHR upon the single qunit state reflects it onto the final state Physical realization of the propagator U requires only two standard Householder reflections

1. Transfer from an arbitrary single state to an arbitrary superposition state 2. Transfer from an arbitrary superposition state to an arbitrary single state 3. Transfer between orthogonal states 4. Transfer between states with real coefficients In several important special cases only a single standard QHR is needed for a pure-to pure transition

NAVIGATION IN HILBERT SPACE Transition by generalized QHR A generalized QHR is ideally for pure-to-pure transition because only one generalized QHR is sufficient to reflect initial state into the final state In this case the solution is unique; there is no arbitrariness in the choice of QHR vector (up to an unimportant global phase) and the phase

COHERENT NAVIGATION BETWEEN MIXED STATES The transfer between two mixed states requires a general U(N) propagator. The latter can be can be expressed as a product of N-1 standard QHRs and one phase gate, or by N generalized QHRs Synthesis of arbitrary preselected mixed state Mixed states with different invariants cannot be connected by Hermitian evolution Because these invariants are constant of motion 1.Using dephasing 2.Using spontaneous emission

Physical Implementation of the coupled Householder reflection HAMILTONIAN OF THE SYSTEM Rabi frequency

Morris-Shore Transformation The system is transformed by Morris-Shore transformation into a set of M independent two-state systems, and a set of K=N-M decoupled dark state

The propagator in the original basis Connects states within the lower set Connects states within the upper set Mix states from the lower and upper sets

COUPLED HOUSEHOLDER REFLECTION Of particular significance is the special case when the Cayley-Klein parameters b n (n=1,2…,M) are all equal to zero All transition probabilities in the MS basis, as well as these in the original basis from the lower set to the upper set, vanish The propagator in the lower sets The propagator in the upper sets

Decomposition of U(N) transformations by Coupled HR Any N-dimensional unitary matrix U can be expressed as a product of one coupled HR and one phase gate

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