Complexity of Gottesman-Kitaev-Preskill States

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.