Efficient classical simulation of cluster state quantum circuits with alternative inputs

We provide new examples of pure entangled systems related to cluster state quantum computation that can be efficiently simulated classically. In cluster state quantum computation input qubits are initialised in the `equator' of the Bloch sphere, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:math> gates are applied, and finally the qubits are measured adaptively using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi></mml:math> measurements or measurements of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>cos</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&amp;#x03B8;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>X</mml:mi><mml:mo>+</mml:mo><mml:mi>sin</mml:mi><mml:mo>&amp;#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>&amp;#x03B8;</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>Y</mml:mi></mml:math> operators. We consider what happens when the initialisation step is modified, and show that for lattices of finite degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math>, there is a constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&amp;#x03BB;</mml:mi><mml:mo>&amp;#x2248;</mml:mo><mml:mn>2.06</mml:mn></mml:math> such that if the qubits are prepared in a state that is within <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&amp;#x03BB;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&amp;#x2212;</mml:mo><mml:mi>D</mml:mi></mml:mrow></mml:msup></mml:math> in trace distance of a state that is diagonal in the computational basis, then the system can be efficiently simulated classically in the sense of sampling from the output distribution within a desired total variation distance. In the square lattice with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> for instance, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>&amp;#x03BB;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>&amp;#x2212;</mml:mo><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.056</mml:mn></mml:math>. We develop a coarse grained version of the argument which increases the size of the classically efficient region. In the case of the square lattice of qubits, the size of the classically simulatable region increases in size to at least around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.070</mml:mn></mml:math>, and in fact probably increases to around <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>&amp;#x2248;</mml:mo><mml:mn>0.1</mml:mn></mml:math>. The results generalise to a broader family of systems, including qudit systems where the interaction is diagonal in the computational basis and the measurements are either in the computational basis or unbiased to it. Potential readers who only want the short version can get much of the intuition from figures 1 to 3.