Quantum recurrent memory stops depending on trainable weights: reset gates make old inputs fade geometrically, with a sensitivity bound flat in sequence length
Synopsis
A quantum recurrent map keeps its memory as a density matrix and implements the forget gate as an input-dependent replacement channel; its output depends on any single input as a trigonometric polynomial whose frequency set is fixed by the encoding, and under a constant gate the nonzero-frequency coefficients decay with that input's age at a rate set by the gate. The sensitivity bound for each input contains the encoding scales, layer count, observable norm and gate values but no trainable unitaries, and it sums uniformly in sequence length; the same weights yield an unbiased estimator built only from reset-free unitary circuits, with expected depth at most (1−fmax)⁻¹.
Interpretation
The recurrent memory state is promoted to a density matrix, and the forget gate becomes an input-dependent replacement channel that swaps the current state for a fixed reference state with probability 1−f, so the state is automatically normalized and the output stays bounded by the observable norm. Earlier quantum recurrent proposals mostly generated gates with variational quantum circuits while keeping the memory itself a classical vector whose norm can grow with the number of steps. In numerical checks, two independent implementations, a direct density-matrix simulator and PennyLane 0.45.1, agree to 9.4×10⁻¹⁶ in trace distance, and normalization and weight-sum deviations sit at the 10⁻¹⁵ level.
The whole recurrence can be written exactly as a convex combination of reset-free unitary trajectories, with weights given by products of gate values, so the output is the weighted average of those trajectory outputs. In the stochastic-resetting literature the state is written as an average over evolution since the last reset, but that treatment used a fixed unitary and a constant reset rate. For n=3, L=2, λ=(1,2,4) and T=8, the largest deviation between the output and the weighted sum over 200 random instances is 5.55×10⁻¹⁶, and 8.05×10⁻¹⁶ in the PennyLane implementation.
The absolute derivative of the output with respect to any input is bounded by a quantity free of the trainable unitaries, containing the product of gate values from that step to the end, the product of encoding-scale sum and layer count, and a gate-derivative term, and the sum over all steps is uniformly bounded in sequence length. A classical LSTM cell admits no bound of this form: scaling every trainable weight by α scales the read-out contribution to input sensitivity by α as well. Sweeping a common weight scale α from 1/4 to 16, the classical cell's maximal total sensitivity rises from 1.2×10⁻² to 4.4×10³, while the map stays below 5.5 against the bound Λ/(1−f)=70; the worst measured-to-bound ratio is 0.347 on random instances and 0.095 for a model fitted to real data by gradient descent.
Perspective
The conclusions concern this class of discrete-time maps: the memory state is a density matrix, the gate is an input-dependent replacement channel, and the step unitary comes from a data re-uploading construction. Numerical verification uses n=3, L=2, λ=(1,2,4), T=8 and 200 random instances; the hardware reconstruction uses T=8, a constant gate f=0.7 and 4096 shots per circuit on ibm fez and ibm kingston. The authors state that at the parameters used the map is efficiently simulable on a classical computer, so the results are structural, describing what this map class can represent and how accurately its output can be estimated from shallow circuits.
The authors state they do not know whether the age-dependent spectral bound or the sensitivity bound is sharp; measured-to-bound ratios of 0.257 and 0.347 on random instances show the bounds hold but are not attained. They also claim no resource advantage over a direct implementation of the channel, because the comparison requiring mid-circuit measurement with conditional reset or an ancilla register was not run. Extending to non-pure reference states and non-unitary step maps changes the unbiased-estimator result, since the trajectory states then become mixed.