The GKP State: A Qubit Drawn as Peak Combs in Light

A Gottesman-Kitaev-Preskill state encodes one qubit as regular peak combs in the continuous amplitudes of a light mode. The ideal combs are perfect and endless. Every real version is a finite-energy approximation.

A comb in each amplitude, not a switch

Most pictures of a qubit look like a two-way switch: spin up or down, charge here or there. A GKP state does something else. It lives in a continuous optical mode and encodes the qubit in where sharp peaks sit along each of two complementary quadratures.

An ideal GKP logical state is a comb of peaks in each quadrature. The two combs complement each other, and each is visible on its own as a one-dimensional projection. A plot along q alone is that projection, not a full two-dimensional lattice in phase space. Ideally the peaks are infinitely sharp and endless, and the state has unbounded energy. No lab can make that object. Real versions use finite squeezing, which spreads each peak into a soft bump, and a Gaussian envelope that cuts the comb off at finite energy. The code can still catch small shifts in those amplitudes.¹

That is a different picture from dual-rail encoding, where one photon in one of two paths is the bit. A GKP qubit is structured light. The same mode can hold many photons on average, rather than one photon in one path.¹²

Why the peaks are hard to make

Preparing even a crude set of peaks needs non-Gaussian light. Photon-number measurements on carefully prepared Gaussian light can herald useful precursors, but a click pattern is not automatically a finished GKP state. Those precursors still need further work, such as multiplexing and breeding, before they look like a usable approximate state.²³

Finite squeezing is the first tax: soft peaks leak probability into the wrong cell. Optical loss is the second, scored by how badly it hurts the quality of the approximate states you can still deliver.²³

A q probability schematic shows finite purple peaks centered on one codeword’s lattice sites, with small coral probability tails extending into neighboring logical decision cells. A separate data-mode line couples to a fresh encoded ancilla, whose measurement supplies a modular q syndrome. A dashed classical arrow controls an inferred conditional displacement; the p step is also required. The data output is not drawn as a sharper comb. Residual error and possibly undetected large-error misassignment remain visible caveats. Conceptual recovery mechanism, not measured performance. — Figure 2: Finite-width q probability peaks have tails that can fall into neighboring logical decision cells. GKP recovery extracts modular q and p syndromes using fresh encoded ancillas and applies a displacement inferred from those outcomes, without measuring the encoded logical value. Small shifts can be corrected, but larger shifts can share the same syndrome and be misassigned. The feedback arrow does not promise narrower peaks, an ideal state, correction of arbitrary loss, or guaranteed recovery. Conceptual schematic, not an experiment.

After the state exists, gates look different

Once approximate GKP modes are available, Clifford operations can be applied deterministically with beamsplitters, phase shifters and homodyne detection, without post-selecting on a successful fusion outcome. Non-Clifford gates still require prepared magic states. These operations can use measurement feed-forward. They cannot compensate for photon loss or for a heralded input that never arrives.

That differs from fusion gates on single-photon dual-rail qubits, which succeed only for some detection outcomes and lean on photon detectors at many stages. The contrast is about the gate step once the light is present, not a claim that GKP has cleared the loss problem. State preparation still depends on heralding, which can fail.²³

What the hardware has shown so far

A June 2025 Nature experiment reported a real optical GKP state from an integrated photonic source, heralded on a (3, 3, 3) photon-number pattern. Both the q and p projections showed at least four resolvable peaks, and the reconstructed Wigner function showed a 3×3 grid of negative regions. Symmetric effective squeezing was 0.62 ± 0.02 dB (0.43 ± 0.02 dB in q, 0.82 ± 0.02 dB in p). Path transmission sat near 78–82%, tomography included that loss with no loss compensation, and the herald rate was about 30 Hz after discarding the noisiest 20% of photon-number outcomes. The same paper simulates that above about 99.5% transmission many higher (n, n, n) patterns with n > 7 could exceed 9.75 dB. That projection was not measured, and the experiment did not demonstrate error suppression.²

Aurora, a 2025 modular photonic machine, is a separate networked-building-blocks demo, not a state-quality benchmark. It networked 35 photonic chips across 4 racks to show heralded non-Gaussian precursors, real-time switching, a large Gaussian cluster state, and adaptive measurements with a live decoder. On the GKP side it used crude two-peak squeezed-cat states, not a dense multi-peak comb, and its refineries did not run the full breeding protocol. The authors say the tested system was too noisy to demonstrate error suppression.³

What the reports leave settled

The idea is older than the hardware: encode a qubit in an oscillator so small amplitude shifts become correctable errors.¹ The integrated source showed a genuine multi-peak optical GKP state at low effective squeezing. The Aurora machine showed modular networking of the surrounding stack. Neither reported experiment demonstrated error suppression.

Two codewords along q compares ideal and finite logical basis codewords of one optical mode. Left: an infinite interleaved comb, with every |0_L> stroke purple and every |1_L> stroke coral, each with the same individual peak spacing and offset by half that spacing. Right: finite-width Gaussian combs on those same centers under a single zero-centered dashed envelope, labeled Finite squeezing, finite energy. Schematic q projection, not full phase space; ideal strokes are symbolic, not measured amplitudes. These are not two physical qubits. — Figure 1: In one q projection, the logical basis codewords are two interleaved combs, offset by half their individual peak spacing; the same states also have structure in the complementary p quadrature. Ideal infinite combs are a mathematical limit; real states have finite-width peaks and finite energy. These are alternative basis codewords of one optical mode, not two physical qubits.

Sources

D. Gottesman, A. Kitaev, and J. Preskill, “Encoding a qubit in an oscillator,” Physical Review A 64, 012310 (2001). DOI: 10.1103/PhysRevA.64.012310. ↩
M. V. Larsen et al., “Integrated photonic source of Gottesman–Kitaev–Preskill qubits,” Nature 642, 587–591 (2025). DOI: 10.1038/s41586-025-09044-5. ↩
H. Aghaee Rad et al., “Scaling and networking a modular photonic quantum computer,” Nature 638, 912–919 (2025). DOI: 10.1038/s41586-024-08406-9. ↩