1-Bit Quantum Filter Overview

Updated 19 January 2026

1-Bit quantum filters are minimal protocols that perform spectral and dynamical filtering using single-qubit resources and simplified measurement records.
They leverage control engineering techniques such as Walsh-function synthesis and quantum invariant filtering to shape noise-filtering properties effectively.
Experimental validations on platforms like trapped ions, NV centers, and circuit QED highlight their ability to reduce error rates and support resource-efficient quantum algorithms.

A 1-bit quantum filter is a quantum information processing protocol, control sequence, or quantum algorithm that achieves spectral, measurement, or dynamical filtering of quantum system evolution using only single-qubit (1-bit) resources, minimal quantum hardware, or minimal classical supervision per iteration. The concept unifies several strands spanning quantum open-system estimation, real-time state purification, robust single-qubit noise filtering, and minimal-resource quantum algorithms for constrained hardware. 1-bit quantum filters are analytically and experimentally well-studied in the context of quantum noise filtering, adaptive quantum algorithms, quantum control engineering, and quantum filtering for state estimation.

1. Theoretical Formulations of 1-Bit Quantum Filters

The term "1-bit quantum filter" encompasses a set of frameworks for quantum filtering problems characterized by the use of a single qubit (or, more strictly, a minimal quantum register) as the filtering element. Key frameworks include:

Noise Filtering in Single-Qubit Gates: The filter-transfer function (FF) framework models the suppression of environmental dephasing or amplitude noise during single-qubit gate operations. For a qubit subject to classical wide-sense-stationary noise during a control operation $U_c(\tau)$ , the average gate infidelity is

$\chi(\tau) = \frac{1}{\pi} \sum_{i \in \{z, \Omega\}} \int_0^\infty d\omega \, S_i(\omega) F_i(\omega)$

where $S_{z,\Omega}(\omega)$ are power spectra and $F_{z,\Omega}(\omega)$ are the corresponding FFs. The "1-bit filter" (filter order $p=1$ ) achieves $F(\omega \tau) \propto (\omega \tau)^2$ for $\omega \tau \ll 1$ , yielding quadratic suppression of low-frequency noise (Soare et al., 2014).

Discrete-time Quantum Filtering with Minimal Record: A continuous observation of a quantum system is compressed to a 1-bit-per-step record (the sign of the measurement increment). The resulting filter update, based only on the sign of measurement, provably supports efficient quantum state estimation and purification, both in theory and in realistic simulations. In the limit of infinitesimal timestep, the 1-bit ("one-bit record," OBR) filter approaches the full stochastic master equation (SME) filter for continuous measurements (Ralph et al., 2011).
Single-ancilla Quantum Algorithms: For quantum algorithms such as ground-state filtering, a "1-bit quantum filter" can refer to the replacement of high-precision quantum phase estimation (QPE) schemes (with $r$ phase-bits) by a single-ancilla ("1-bit") threshold circuit. This is utilized in near-term sparse Hamiltonian filtering where only a binary decision is needed to separate signal from combinatorial background, as in the TrackHHL algorithm (Chiotopoulos et al., 12 Jan 2026).
Quantum Invariant Filtering (QIF): QIF provides an optimal protocol-based design prescription: given a desired frequency-domain filter transfer function, the method yields the continuous single-qubit driving field that realizes the filter at the quantum level, supporting arbitrary impulse responses including single/multi-band and phase-sensitive filters (Cangemi et al., 18 Jun 2025).

2. Control Engineering and Spectral Characterization

1-bit quantum filters are operationalized through analytic and numerical control engineering methods that synthesize filter properties in the single-qubit (or minimal bit) control manifold:

Walsh-function Basis and Hadamard Construction: Quantum filtering via Walsh bases constructs piecewise-constant amplitude profiles using a signed sum of orthonormal binary-valued (Walsh) functions. For a control window of duration $\tau$ with $M=4$ segments, the control profile is

$\Omega(t) = X_0 \, {\rm PAL}_0(t/\tau) + X_3 \, {\rm PAL}_3(t/\tau)$

so that the FF $F_z(\omega)$ can be shaped to first order ( $p=1$ ) low-pass suppression; only one Walsh weight ( $X_3$ ) is varied in optimization (Soare et al., 2014).

Filter Order and Scaling: The low-frequency scaling of the filter-transfer function is dictated by the filter order $p$ : $F(\omega \tau) \propto (\omega \tau)^{2p}$ near $\omega=0$ . Doubling the number of segments increases achievable order by one ( $M=8 \implies p=2$ ) (Soare et al., 2014, Green et al., 2011).
Continuous Protocol Synthesis (QIF): QIF maps a designed classical finite-impulse response (FIR) transferring function $H(\omega)$ into a quantum time-dependent driving field $H(t)$ via a dynamical invariant construction, e.g., by setting the auxiliary variable $\beta(t) = -\pi/2 + \arcsin[H(t)]$ and extracting the drive amplitude $\varepsilon(t) = \partial_t \beta(t)$ . This preserves frequency-domain behavior at the quantum level (Cangemi et al., 18 Jun 2025).
Minimal-resource Threshold Algorithms: In quantum algorithms for combinatorial optimization, imposing a 1-bit threshold on spectral phase estimation (e.g., in the TrackHHL scheme) yields asymptotic gate-depth scaling $O(\sqrt{N} \log N)$ , compared to $O(\kappa \, {\rm polylog}(N))$ in full HHL, for $N$ -dimensional, $k$ -sparse Hamiltonians (Chiotopoulos et al., 12 Jan 2026).

3. Algorithmic and Measurement-based Quantum Filtering

Quantum filters in the "1-bit" paradigm may be implemented algorithmically in both continuous-measurement and circuit-model settings:

Stochastic Master Equations with 1-bit Records: For a weakly measured qubit, the continuous SME can be discretized such that at each timestep the measurement record is digitized to a single bit (only the sign of the measurement increment is retained). The update is performed via a simple Kraus-operator map. For measurement strength $\kappa$ ,

$\rho_{n+1} = \frac{M_{y_n} \, \rho_n \, M_{y_n}^\dagger}{\mathrm{Tr}(M_{y_n} \, \rho_n \, M_{y_n}^\dagger)}$

with $M_{y_n} = \sqrt{\frac{1}{2}}\left(I + y_n \sqrt{8\kappa \Delta t}\, y\right)$ . The filter converges to the full analog solution as $\Delta t \rightarrow 0$ (Ralph et al., 2011).

Ground-state Filtering via Single-ancilla Circuits: In TrackHHL, each possible track segment in a collider tracking problem maps to a binary register, and the Ising-like Hamiltonian admits a sparse representation. A single phase-bit is used to apply a filter at spectral threshold; a measurement of an ancilla post-selects eigenstates that satisfy the ground-state filter (below or above the cutoff) (Chiotopoulos et al., 12 Jan 2026).
Feedback and Purification: Using 1-bit filtered measurement records, rapid purification and real-time feedback are achievable—even with severe classical data compression—by applying simple unitary feedback (e.g., rotating the Bloch vector onto the measurement axis). Purification rates approach those of full analog-record filters (Ralph et al., 2011).
Open Quantum Systems with Non-Markovianity: In non-Markovian continuous quantum filtering, "1-bit" refers to the tracking of a single qubit's conditional quantum state under arbitrary colored-noise (arbitrary spectrum) via a pseudo-mode (multi-ancilla) model, with the estimator itself remaining 1-bit in the principal system (Xue et al., 2015).

4. Experimental Realizations and Performance Benchmarks

1-bit quantum filters have been validated experimentally across diverse physical platforms:

Trapped Ions: Piecewise-constant, Walsh-synthesized 1-bit filters, with only four control segments per $\pi$ -gate, were realized in $^{171}$ Yb $^+$ ion chains. Fidelity suppression against engineered dephasing noise was observed, with gate error reduced from $\sim0.8\%$ (primitive pulse) to $\sim0.3\%$ (1-bit W1 filter) in non-Markovian baths. The measured frequency-rolloff followed $F_z(\omega\tau) \propto (\omega\tau)^2$ up to $\omega\tau \sim 0.3$ , as predicted (Soare et al., 2014).
Diamond NV Centers: The QIF protocol was implemented via continuous amplitude modulation on the $\left|0\right\rangle \leftrightarrow \left|-1\right\rangle$ spin transition, achieving bandpass/selective coherence filtering and phase-sensitive quantum lock-in detection. QIF extended coherence time beyond Carr-Purcell-Meiboom-Gill by two orders of magnitude and remained robust to $\pm 50\%$ drive amplitude variations (Cangemi et al., 18 Jun 2025).
Circuit QED (Saturable Purcell Filter): A single "filter qubit" (Josephson quantum filter, JQF) in the transmission line between measurement resonator and environment suppresses qubit Purcell decay. The filter is "switched off" (becomes transparent) when saturated by a strong control pulse, maintaining high-fidelity single-qubit operations and compatibility with frequency-multiplexed architectures (Iakoupov et al., 2022).
NISQ-era Benchmarking of Quantum Algorithms: The 1-bit quantum filter (TrackHHL) demonstrated feasible scaling on hardware models: gate complexity scaling as $O(\sqrt{N} \log N)$ , success probability $P_{\mathrm{succ}} \sim N^{-1/2}$ , and superior signal separation on all-to-all connectivity trapped-ion processors (Quantinuum H2) compared to superconducting (IBM Heron) platforms. The approach remains practical for up to $\sim 20$ binary variables (segment candidates) on current devices (Chiotopoulos et al., 12 Jan 2026).

5. Extensions, Design Principles, and Generalizations

The 1-bit quantum filter paradigm demonstrates adaptability and extension to multiple quantum information processing contexts:

Higher-order Filtering: By increasing the number of segments or controls (e.g., $M=8$ segments in the Walsh basis), higher-order filtering ( $p=2$ , $F \propto (\omega\tau)^4$ ) is achievable (Soare et al., 2014).
Platform Agnosticism: The synthetic design protocols underlying 1-bit filters do not depend on details of the quantum platform, and are widely applicable to trapped ions, NV centers, superconducting qubits (flux or microwave control), and architectures supporting high-fidelity arbitrary single-qubit control (Soare et al., 2014, Cangemi et al., 18 Jun 2025, Iakoupov et al., 2022).
Algorithmic Generalization: The single-ancilla, 1-bit thresholding approach is general in quantum algorithms where the problem reduces to binary eigenvalue discrimination, making it especially suitable for sparse Hamiltonians and binary optimization problems (Chiotopoulos et al., 12 Jan 2026).
Filter Customization: The QIF approach directly maps any classical FIR filter specification (impulse response, bandpass/bandstop, multi-band, or phase-lock) to a feasible quantum protocol, providing maximal design flexibility (Cangemi et al., 18 Jun 2025).
Multi-qubit Extension and Multiplexing: 1-bit Walsh filters can be embedded in two-qubit gate control (e.g., Mølmer–Sørensen entangling gates), and saturable Purcell filters can be frequency-multiplexed for scalable readout with minimal transmission lines (Soare et al., 2014, Iakoupov et al., 2022).

6. Impact, Limitations, and Outlook

1-bit quantum filters have established a principled connection between quantum control, noise-suppression, real-time estimation, and resource-efficient quantum algorithms:

Impact: They offer quantitative, platform-independent filter-design toolkits with rigorous performance predictions. Experimental validations confirm substantial error suppression, spectral flexibility, and resource efficiency of both pulsed and continuous implementations (Soare et al., 2014, Cangemi et al., 18 Jun 2025, Chiotopoulos et al., 12 Jan 2026).
Limitations: The principal limitation is in success probability and scaling: for $N$ -dimensional problems (e.g., TrackHHL), $P_\mathrm{succ} \sim 1/\sqrt{N}$ , requiring repetition or amplitude amplification for large $N$ . In feedback/estimation settings, finite sampling rate and measurement inefficiency slow purification. In noise filtering, filter order is limited by segment count or smoothness constraints.
Outlook: The methodology paves the way toward noise-optimized, resource-minimal gate synthesis, scalable measurement protocols, and domain-specific quantum filtering algorithms compatible with the constraints of NISQ hardware. Ongoing work generalizes 1-bit filter schemes to multi-qubit open systems, dynamical decoupling, and analog/digital hybrid protocols.