Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations
Anurag Anshu, Tony Metger
Abstract
We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from \cite{DMRF22}; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>e</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>&#x03B9;</mml:mi><mml:msup><mml:mi>H</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mo>&#x22EF;</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>&#x03B9;</mml:mi><mml:msup><mml:mi>H</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:msub><mml:mi>&#x03C8;</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo></mml:math> for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math>-qubit product state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:msub><mml:mi>&#x03C8;</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo fence="false" stretchy="false">&#x27E9;</mml:mo></mml:math>, where each <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>i</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math> can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approximations to show that these states are close to local operators. This implies that the distribution of the Hamming weight of a computational basis measurement (and of other related observables) concentrates.An example of (iii) are the states produced by the quantum approximate optimisation algorithm (QAOA). Using our concentration results for these states, we show that for a random spin model, the QAOA can only succeed with negligible probability even at super-constant level <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, assuming a strengthened version of the so-called overlap gap property. This gives the first limitations on the QAOA on dense instances at super-constant level, improving upon the recent result [BGMZ22].