Algorithmic Shadow Spectroscopy

We present shadow spectroscopy as a simulator-agnostic quantum algorithm for estimating energy gaps using very few circuit repetitions (shots) and no extra resources (ancilla qubits) beyond performing time evolution and measurements. The approach builds on the fundamental feature that every observable property of a quantum system must evolve according to the same harmonic components: we can reveal them by postprocessing classical shadows of time-evolved quantum states to extract a large number of time-periodic signals <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:msub><a:mi>N</a:mi><a:mi>o</a:mi></a:msub><a:mo>∝</a:mo><a:msup><a:mn>10</a:mn><a:mn>8</a:mn></a:msup></a:math>, whose frequencies correspond to Hamiltonian energy differences with precision limited as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mi>ϵ</d:mi><d:mo>∝</d:mo><d:mn>1</d:mn><d:mo>/</d:mo><d:mi>T</d:mi></d:math> for simulation time <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mi>T</g:mi></g:math>. We provide strong analytical guarantees that (a) quantum resources scale as <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><j:mi>O</j:mi><j:mo stretchy="false">(</j:mo><j:mi>log</j:mi><j:mo></j:mo><j:msub><j:mi>N</j:mi><j:mi>o</j:mi></j:msub><j:mo stretchy="false">)</j:mo></j:math>, while the classical computational complexity is linear <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><o:mi>O</o:mi><o:mo stretchy="false">(</o:mo><o:msub><o:mi>N</o:mi><o:mi>o</o:mi></o:msub><o:mo stretchy="false">)</o:mo></o:math>, (b) the signal-to-noise ratio increases with the number of processed signals as <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><t:mo>∝</t:mo><t:msqrt><t:msub><t:mi>N</t:mi><t:mi>o</t:mi></t:msub></t:msqrt></t:math>, and (c) spectral peak positions are immune to reasonable levels of noise. We demonstrate our approach on model spin systems and the excited-state conical intersection of molecular <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><w:msub><w:mi>CH</w:mi><w:mn>2</w:mn></w:msub></w:math> and verify that our method is indeed intuitively easy to use in practice, robust against gate noise, amiable to a new type of algorithmic-error mitigation technique, and uses relatively few shots given a reasonable initial state is supplied—we demonstrate that even 10 shots per time step can be sufficient. Finally, we measured a high-quality, experimental shadow spectrum of a spin chain on readily available IBM quantum computers, achieving the same precision as in noise-free simulations without using any advanced error mitigation, and verified scalability in tensor-network simulations of up to 100-qubit systems.