Towards scalable multi-qubit optimal control via interaction decomposition in the diagonal frame

Abstract

In this work, we introduce a general n-qubit formulation of control objectives that allows a control target to be specified in a diagonal frame, so that only the diagonal entries must be characterized, thus quadratically reducing the complexity of the cost functional in constrast to a full target matrix. We do so by representing any n-qubit unitary transformation as a diagonal phase map on the computational basis states, as they are naturally diagonalizable by unitarity. By using discrete derivative operators to analytically construct support-selective phase invariants, we enable to deterministically isolate and quantify any multi-qubit interactions encoded in the phase map. These phase invariants form a coordinate system for the formulation of specific control targets in terms of arbitrary desired multi-qubit interactions, without having to invert the diagonalization during the optimizatiion, solely relying on the experimentally accesible diagonal phases. To illustrate the framework, we synthesize two genuinely tripartite entangling gates, both, diagonal and non-diagonal. These are obtained with a single shaped microwave pulse, for a numerically simulated room-temperature nitrogen-vacancy center with a three qubit nuclear spin register, with durations of about a microsecond. These results represent a factor 10-100 reduction in operation time compared with the fastest existing NV-based entanglers that act on more than two qubits at once.

Figures (6)

• Figure 1: Optimized microwave envelope realizing the diagonal phase-map control objective. The pulse selectively results in a finite three-body phase as given by the three-body phase invariant $\Delta_{\{a,b,c\}}\approx\pi/4$ modulo $\pi$, while suppressing all two-body phases as given by the two-body correlators $\Delta_{\{a,b\}},\Delta_{\{a,c\}},\Delta_{\{b,c\}}$ similarly.
• Figure 2: Time evolution of the phase invariants $\Delta_S(\phi(t))$ extracted from the diagonal phase map of the logical three-qubit subspace during the optimized pulse. To probe all phases specified with the given set of phase invariants, we initialize the system in $\ket{+}$ states in the diagonalizing frame. The three-body phase invariant $\Delta_{\{a,b,c\}}$ (thick black line) converges to $-3\pi/4$, which is physically equivalent to the target value $\pi/4$ in entangling effect. The two-body phase invariants $\Delta_{\{a,b\}}$, $\Delta_{\{a,c\}}$, and $\Delta_{\{b,c\}}$ approach integer multiples of $\pi$, corresponding to vanishing pairwise interaction content. Single-qubit phase invariants $\Delta_{\{a\}},\Delta_{\{b\}},\Delta_{\{c\}}$ correspond to local $Z$ phases.
• Figure 3: Three-qubit population dynamics during the optimized control pulse for implementing the diagonal target gate $e^{\,i\frac{\pi}{4}Z\!\otimes\!Z\!\otimes\!Z}$. The shaded area indicates the time-integrated population of the decoherence-prone electronic $m_s=-1$ manifold used to estimate the accumulated dephasing.
• Figure 4: Optimized microwave envelope implementing the non-diagonal three-qubit entangler $e^{i(\pi/4)X\!\otimes\!Z\!\otimes\!Z}$ in the computational basis, synthesized via phase invariant optimization in the diagonalizing (Hadamard) frame.
• Figure 5: Time evolution of the phase invariants $\Delta_S(\phi(t))$ extracted from the diagonal phase map in the diagonalizing (Hadamard) frame during the optimized non-diagonal gate pulse. To be able to probe all phases that we specified via phase invariants, we initialize the system in $\ket{+}$ states in the diagonalizing frame, corresponding to the computational-basis state $\ket{0++}$. As in Fig. \ref{['fig:nv_3q_invariants_vs_time_ZZZ']}, the three-body phase invariants $\Delta_{\{a,b,c\}}$ (thick black line) converges to $\pi/4$, realizing the desired three-body interaction, up to single-qubit phases, in the Hadamard frame. The two-body phase invariants $\Delta_{\{a,b\}}$, $\Delta_{\{a,c\}}$, and $\Delta_{\{b,c\}}$ approach values corresponding to vanishing pairwise interaction content.

Theorems & Definitions (6)

• Proposition 1: Support Inclusion Filtering
• Remark 1
• Proposition 2: support-selective phase invariants
• Definition 1: Support-Selective Phase Invariant based Control Objective (Three Qubits)
• Lemma 1: Alternating-sum Identity of phase invariants
• proof