Complexity of Gottesman-Kitaev-Preskill States
Bibliographic record
Abstract
We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations and single- and two-qubit operations as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an approximation to the state in trace distance. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mo stretchy="false">|</a:mo> <a:msub> <a:mrow> <a:mi mathvariant="sans-serif">G</a:mi> <a:mi mathvariant="sans-serif">K</a:mi> <a:mi mathvariant="sans-serif">P</a:mi> </a:mrow> <a:mrow> <a:mi>κ</a:mi> <a:mo>,</a:mo> <a:mi mathvariant="normal">Δ</a:mi> </a:mrow> </a:msub> <a:mo stretchy="false">⟩</a:mo> </a:math> . Here, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>κ</i:mi> <i:mrow> <i:mo>−</i:mo> <i:mn>2</i:mn> </i:mrow> </i:msup> </i:math> is the variance of the envelope, and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mi mathvariant="normal">Δ</k:mi> <k:mn>2</k:mn> </k:msup> </k:math> is the variance of the individual peaks. We show that the circuit accepts with constant probability and—conditioned on acceptance—the output state is polynomially close in <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mo stretchy="false">(</n:mo> <n:mi>κ</n:mi> <n:mo>,</n:mo> <n:mi mathvariant="normal">Δ</n:mi> <n:mo stretchy="false">)</n:mo> </n:math> to the state <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mo stretchy="false">|</s:mo> <s:msub> <s:mrow> <s:mi mathvariant="sans-serif">G</s:mi> <s:mi mathvariant="sans-serif">K</s:mi> <s:mi mathvariant="sans-serif">P</s:mi> </s:mrow> <s:mrow> <s:mi>κ</s:mi> <s:mo>,</s:mo> <s:mi mathvariant="normal">Δ</s:mi> </s:mrow> </s:msub> <s:mo stretchy="false">⟩</s:mo> </s:math> . The size of our circuit is linear in <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mo stretchy="false">(</ab:mo> <ab:mi>log</ab:mi> <ab:mn>1</ab:mn> <ab:mo>/</ab:mo> <ab:mi>κ</ab:mi> <ab:mo>,</ab:mo> <ab:mi>log</ab:mi> <ab:mn>1</ab:mn> <ab:mo>/</ab:mo> <ab:mi mathvariant="normal">Δ</ab:mi> <ab:mo stretchy="false">)</ab:mo> </ab:math> . To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".