Dynamical decoupling and homogenization of continuous variable systems

Abstract

Abstract. For finite-dimensional quantum systems, such as qubits, a well established strategy to protect such systems from decoherence is dynamical decoupling. However many promising quantum devices, such as oscillators, are infinite dimensional, for which the question if dynamical decoupling could be applied remained open. Here we first show that not every infinite-dimensional system can be protected from decoherence through dynamical decoupling. Then we develop dynamical decoupling for continuous variable systems which are described by quadratic Hamiltonians. We identify a condition and a set of operations that allow us to map a set of interacting harmonic oscillators onto a set of non-interacting oscillators rotating with an averaged frequency, a procedure we call homogenization. Furthermore we show that every quadratic system-environment interaction can be suppressed with two simple operations acting only on the system. Using a random dynamical decoupling or homogenization scheme, we develop bounds that characterize how fast we have to work in order to achieve the desired uncoupled dynamics. This allows us to identify how well homogenization can be achieved and decoherence can be suppressed in continuous variable systems.

An upper bound on the time required to implement unitary operations

“…Clearly, N is upper bounded by the total number of vertices present in G 0 , which is given by the dimension d of the quantum system. Introducing the smallest edge weight present in G 0 as g min = min (n,m)∈E 0 {|g n,m |} we find that the time t n,m to implement a generic S n,m (α) with (n, m) ∈ E K is therefore upper bounded by (12).…”

“…In order to analyze the tightness of the obtained bounds, we compare the bounds (12) and (16) to previously derived lower bounds [15], as well as to minimum gate times obtained from numerical gate optimization using the GRAPE algorithm [16], which is included in the Python package QuTip [17]. Similar to the method utilized in [15], a population binary search algorithm is run over T until the gate error is smaller than 10 −4 .…”

“…We proceed by considering random drift Hamiltonians that correspond to random connected graphs. Throughout this section the couplings |g n,m | are chosen to be uniformly random in the interval [1,2] so that g min = 1 and we study the validity of the bounds (12) and (16) for quantum systems of dimension d ∈ [2,6], noting that for a fixed d ∈ [2,6] we have {1, 2, 6, 21, 112} distinct connected graph.…”

Amplification of quadratic Hamiltonians

“…(2)) can be amplified through rapidly applied local parametric controls. In contrast, for some infinite dimensional systems the opposite, i.e., Dynamical Decoupling (DD) that achieves averaging out Hamiltonians, is impossible [11], while for finite dimensional systems DD can always succeed [12]. However, for infinite dimensional systems DD can be used to suppress certain interactions.…”

Universal Uhrig Dynamical Decoupling for Bosonic Systems

“…A similar analysis applies to (m + 1)-qubit DD schemes with multi-qubit Pauli pulses, see [13]: Identically defined functions (16) appear in the toggling frame generator and give rise to the same coefficients (18) in the Dyson series of the toggling frame evolution. As a consequence, any qubit decoupling scheme based on Pauli pulses provides the necessary cancellations when translated to the bosonic homogenization setting using the substitution rule (11).…”