Statement
A qubit coupled to a bath evolves under the most general Hamiltonian
$$H = I \otimes B_I + X \otimes B_X + Y \otimes B_Y + Z \otimes B_Z,$$
with arbitrary bounded bath operators \(B_\alpha\) (\(B_I\) is the bath’s own Hamiltonian). A dynamical decoupling sequence applies ideal, instantaneous \(\pi\) pulses about the \(x\) and \(z\) axes at chosen times in \([0, T]\). It achieves order \(K\) if the resulting evolution, in the toggling frame, contains no term that couples qubit and bath up to order \(T^K\) — for every choice of the \(B_\alpha\), and to all orders in the coupling strength [1, 2].
Let \(N(K)\) be the smallest number of pulses that achieves order \(K\).
Question. How does \(N(K)\) grow? Is \(N(K) = \Theta(K^2)\), as the quadratic construction suggests, or is \(N(K) = O(K)\)?
Why it matters
Dynamical decoupling is the cheapest way to protect idle qubits and is used routinely on current processors. Each pulse costs time and adds pulse errors, so the number of pulses needed for a given order of protection decides how far high-order schemes are useful in practice. The question is also a clean test of what optimal control can achieve with the simplest possible controls: a finite set of instantaneous rotations.
Known results
One error axis. For pure dephasing, or for pure relaxation, Uhrig’s sequence reaches order \(K\) with \(K\) pulses [3], and no sequence does it with fewer [4, 1]. Since general noise contains these cases, \(N(K) \ge K\); a self-contained proof of this lower bound is given in [5].
Quadratic dynamical decoupling (QDD). Nesting a Uhrig sequence about one axis inside a Uhrig sequence about an orthogonal axis cancels general decoherence to order \(K\) with \((K+1)^2\) pulse intervals, so \(N(K) = O(K^2)\) [1]. Concatenated decoupling, the first scheme to reach arbitrary order, needs a number of pulses that grows exponentially with \(K\) [6, 5]. The universality of QDD was proved first for even inner order [7] and then for all inner and outer orders [2], with rigorous performance bounds for bounded baths [8].
Small orders. Exhaustive numerical searches give optimal sequences of 7, 14 and 23 pulses for orders 2, 3 and 4 — that is, \((K+1)^2 - 2\), differing from QDD by at most two pulses [1].
Weak coupling. If only the terms linear in the coupling strength \(J\) must vanish through order \(K\), then \(O(K)\) pulses suffice — at most \(3K\) for a single qubit — and linear scaling is optimal. The construction maps pulse scheduling to the continuous necklace-splitting problem. Its error is \(O(JT^{K+1}) + O(J^2T^2)\), so it beats QDD when the coupling is weak; outside that regime QDD, which also cancels higher orders in \(J\), can perform better [5].
Status
Open. For decoupling to all orders in the coupling, the best known sequences use \(\Theta(K^2)\) pulses and the best lower bound is \(K\). The small-order searches and the conjecture of near-optimality point to quadratic growth, but no lower bound beyond linear is known, and we have not found a construction with \(o(K^2)\) pulses. If you know one, please tell us. The asymptotic form of the question is our formulation of the near-optimality conjecture of West, Fong and Lidar.