Nonadiabatic holonomic quantum computation with atom-cavity system

<p indent="0mm">Quantum computation is much more effective than classical computation in solving many problems such as factoring large integers and searching unsorted databases. However, such effectiveness relies on the ability to perform universal high-fidelity quantum gates. One main challenge in achieving such high-fidelity gates is to reduce control errors of a quantum system. To overcome this problem, various proposals of fault-tolerant quantum computation are proposed. A promising one of such proposals is nonadiabatic holonomic quantum computation. Nonadiabatic holonomic quantum computation is realized by using a quantum system with a subspace satisfying both the cyclic evolution and parallel transport conditions. For an <italic>N</italic> dimensional quantum system with Hamiltonian <italic>H</italic>(<italic>t</italic>) and evolution operator <inline-formula id="INLINE98"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>T</mml:mi><mml:mi>exp</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mo>−</mml:mo><mml:mtext>i</mml:mtext><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo other="0">∫</mml:mo><mml:mn other="1">0</mml:mn><mml:mi other="1">t</mml:mi></mml:msubsup><mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo>'</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mtext>d</mml:mtext><mml:mi>t</mml:mi><mml:mo>'</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mo stretchy="false">]</mml:mo></mml:mrow></mml:math></inline-formula>, if there exists a time-dependent <italic>L</italic> dimensional subspace <italic>S</italic>(<italic>t</italic>) spanned by the orthonormal vectors <inline-formula id="INLINE99"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow other="0"><mml:mo other="0">{</mml:mo><mml:mrow other="0"><mml:mo other="0">|</mml:mo><mml:mrow other="0"><mml:msub other="0"><mml:mi other="0">ϕ</mml:mi><mml:mi other="1">k</mml:mi></mml:msub><mml:mo other="0" stretchy="false">(</mml:mo><mml:mi other="0">t</mml:mi><mml:mo other="0" stretchy="false">)</mml:mo></mml:mrow><mml:mo other="0">⟩</mml:mo></mml:mrow><mml:mo other="0">=</mml:mo><mml:mi other="0">U</mml:mi><mml:mo other="0" stretchy="false">(</mml:mo><mml:mi other="0">t</mml:mi><mml:mo other="0" stretchy="false">)</mml:mo><mml:mrow other="0"><mml:mo other="0">|</mml:mo><mml:mrow other="0"><mml:msub other="0"><mml:mi other="0">ϕ</mml:mi><mml:mi other="1">k</mml:mi></mml:msub><mml:mo other="0" stretchy="false">(</mml:mo><mml:mn other="0">0</mml:mn><mml:mo other="0" stretchy="false">)</mml:mo></mml:mrow><mml:mo other="0">⟩</mml:mo></mml:mrow><mml:mo other="0">}</mml:mo></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi><mml:mo other="1">=</mml:mo><mml:mn other="1">1</mml:mn></mml:mrow><mml:mi other="1">L</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula> that satisfy the two conditions: (1) <inline-formula id="INLINE100"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mrow other="0"><mml:mo other="0">∑</mml:mo></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi><mml:mo other="1">=</mml:mo><mml:mn other="1">1</mml:mn></mml:mrow><mml:mrow other="1"><mml:mi other="1">L</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mrow other="0"><mml:mi other="0">ϕ</mml:mi></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>⟩</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mrow><mml:mo>⟨</mml:mo><mml:mrow><mml:msub><mml:mrow other="0"><mml:mi other="0">ϕ</mml:mi></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo></mml:mrow></mml:math></inline-formula> <inline-formula id="INLINE101"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mrow other="0"><mml:mo other="0">∑</mml:mo></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi><mml:mo other="1">=</mml:mo><mml:mn other="1">1</mml:mn></mml:mrow><mml:mrow other="1"><mml:mi other="1">L</mml:mi></mml:mrow></mml:msubsup><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mrow other="0"><mml:mi other="0">ϕ</mml:mi></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>⟩</mml:mo></mml:mrow></mml:mrow></mml:mstyle><mml:mrow><mml:mo>⟨</mml:mo><mml:mrow><mml:msub><mml:mrow other="0"><mml:mi other="0">ϕ</mml:mi></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, with <inline-formula id="INLINE102"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi></mml:math></inline-formula> being the evolution period, and (2) <inline-formula id="INLINE103"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mrow><mml:mo>⟨</mml:mo><mml:mrow><mml:msub><mml:mi other="0">ϕ</mml:mi><mml:mi other="1">k</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:mi>H</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mrow><mml:mo>|</mml:mo><mml:mrow><mml:msub><mml:mi other="0">ϕ</mml:mi><mml:mi other="1">l</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>⟩</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math></inline-formula><inline-formula id="INLINE104"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mo>,</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:math></inline-formula> then the unitary transformation <inline-formula id="INLINE105"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> is a holonomic gate on the <italic>L</italic> dimensional subspace <italic>S</italic>(0) spanned by <inline-formula id="INLINE106"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow other="0"><mml:mo other="0">{</mml:mo><mml:mrow other="0"><mml:mo other="0">|</mml:mo><mml:mrow other="0"><mml:msub other="0"><mml:mi other="0">ϕ</mml:mi><mml:mi other="1">k</mml:mi></mml:msub><mml:mo other="0" stretchy="false">(</mml:mo><mml:mn other="0">0</mml:mn><mml:mo other="0" stretchy="false">)</mml:mo></mml:mrow><mml:mo other="0">⟩</mml:mo></mml:mrow><mml:mo other="0">}</mml:mo></mml:mrow><mml:mrow other="1"><mml:mi other="1">k</mml:mi><mml:mo other="1">=</mml:mo><mml:mn other="1">1</mml:mn></mml:mrow><mml:mi other="1">L</mml:mi></mml:msubsup></mml:mrow></mml:math></inline-formula>. The work of realizing nonadiabatic holonomic gate is to build a physical system with a Hamiltonian satisfying both the cyclic evolution condition and the parallel transport condition. Nonadiabatic holonomic quantum computation is based on nonadiabatic non-Abelian geometric phases. Since nonadiabatic non-Abelian geometric phases are only dependent on evolution paths but independent of evolution details, nonadiabatic holonomic gates are robust against control errors. Besides, nonadiabatic holonomic gates do not require the long run-time evolution that is necessary for adiabatic holonomic gates. Due to the merits of both robustness against control errors and high-speed realization, nonadiabatic holonomic quantum computation has received increasing attention in theory and experiment. Various schemes of nonadiabatic holonomic quantum computation have been proposed based on different physical systems, and a number of these schemes have been experimentally demonstrated with nuclear magnetic resonance systems, nitrogen-vacancy center systems, and superconducting systems. The progress of nonadiabatic holonomic quantum computation is also reflected in the improvement of the methods of designing quantum gates. The single-shot scheme, single-loop multiple-pulse scheme and composite gate scheme of nonadiabatic quantum gate are based on the improved methods. In particular, the single-shot scheme simplifies the operation of the one-qubit gate, so that the operation of any one-qubit gate, which needs to be realized by combining two non-commuting π-rotation gate in the original scheme, can be simply realized by a single-shot operation. In this paper, we propose a scheme of nonadiabatic holonomic quantum computation with atom-cavity system. The universal set of quantum gates in our scheme consists of an arbitrary one-qubit nonadiabatic holonomic gate and a two-qubit nonadiabatic holonomic controlled-PHASE gate. To realize the arbitrary one-qubit gate, we use two resonant laser pulses and an off-resonant laser pulse to drive a single four-level atom. With the aid of Stark shifts, the one-qubit gate can be performed in one-step only by changing the Rabi frequencies and the phases of the laser pulses. To realize the two-qubit controlled-PHASE gate, we use two off-resonant laser pulses to drive the two four-level atoms in the cavity. The transitions between the two-atom ground state and the two-atom auxiliary state is facilitated by exchanging virtual photons through a common cavity mode. In this process, the effective two-atom transition is disentangled from the cavity mode and therefore the two-qubit gate is insensitive to the cavity decay.