Deterministic Bethe state preparation
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Abstract
<jats:p>We present an explicit quantum circuit that prepares an arbitrary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>-eigenstate on a quantum computer, including the exact eigenstates of the spin-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>2</mml:mn><mml:mi>X</mml:mi><mml:mi>X</mml:mi><mml:mi>Z</mml:mi></mml:math> quantum spin chain with either open or closed boundary conditions. The algorithm is deterministic, does not require ancillary qubits, and does not require QR decompositions. The circuit prepares such an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>-qubit state with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>M</mml:mi></mml:math> down-spins using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow class="MJX-TeXAtom-OPEN"><mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo></mml:mrow><mml:mfrac linethickness="0"><mml:mi>L</mml:mi><mml:mi>M</mml:mi></mml:mfrac><mml:mrow class="MJX-TeXAtom-CLOSE"><mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo></mml:mrow></mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math> multi-controlled rotation gates and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mi>M</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>L</mml:mi><mml:mo>−</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> CNOT-gates.</jats:p>