Error analysis of gate-based quantum computers with transmon qubits Dennis Willsch1, Madita Nocon1, Fengping Jin1, Hans De Raedt2, Kristel Michielsen1,3 1 Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, D-5 ! 5 Jülich, "erman# 2 $erni%e Institute for Advanced &aterials, University of "roningen, (i)en*orgh !, (+-,-!7 A" "roningen, .he (etherlands 3 /0.H Aachen Universit#, D-5 256 Aachen, "erman# Gate-based quantum computing Transmon qubit architecture A quantum computer containsj i a setj i of two-level systems called !he architecture of the transmon quantum computer is defined by the system Hamiltonian qubits. Each qubit can be in a complex superposition of the computational states 0 and 1 . At each step in the !he qubits are given by the lowest eigenstates of computation, gates transform the qubits. 4ooper Pair Boxes %4$Bs) in the transmon regime [1]: Examples for single-qubit gates 6ne way of coupling transmons is based on a trans- mission line resonator, modeled as a harmonic oscillator Another way of coupling transmons is based on a The two-qubit controlled-NOT capacitive electrostatic interaction: %456!& gate is a conditional operation to entangle two qubits. !he computation can be expressed as a quantum circuit 9uantum gates are implemented by microwave voltage pulses: j i j i At the end, a measurement of the qubits produces a bit string by pro"ecting each qubit to 0 or 1Simulation. method Gate-error metrics t M U U !he time-dependent Schr;dinger The 456! gate is implemented in three different Projection of the time-evolution operator % & on the qubit equation %!+SE) versions based on cross-resonance %47& pulses (0*. subspace gives the matrix . 'deally, this matrix should be equal to the unitary quantum gate . is solved numerically using a Su/u<i- !rotter product-formula algorithm (=* for the time-evolution operator Average gate fidelity ()* n t gi +iamond error rate (,* t U !he goal is to find a pulse % & so that % & implements a certain quantum gate on the qubits. We use Unitarity (.* the Nelder-Mead algorithm to optimize the parameters of the pulse. Simulation results References: [1] J. Koch et al. Ph#s5 /ev5 A 7 , !2319 (20 $% [2] H. De Raedt. Comp5 Ph#s5 /ep5 7, 1 #19&$% [3] '. 'heldon et al. Ph#s5 /ev5 A5 !", (03 2 (201(% [!] M. Nielsen. 4h#s5 +ett5 A5 "#", 2!9 (20 2% [)] *. Kitaev. /ussian &athematical Surveys $%, 1191 (1""$% [(] J. Wallman et al. (ew J5 Ph#s5 17, 11302 (201)% [$] K. Michielsen et al. Comp5 4h#s5 Comm. %%#, !! (201$% [&] -.M /0ant0, 12perience. https344555.research.ib,.com4ib,78 (2 1(% ["] D. Willsch et al. Ph#s5 /ev5 A ! , (23 2 (201$% Conclusion: !he gate metrics of the optimized pulses are nearly perfect and agree with experimental achievements (0]. However, in repeated applications or actual quantum circuits, the gates suffer from systematic errors. These can be observed in experiments (1,8]. Although the gate fidelity and other metrics indicate them, they cannot replace the information of how well and how often a certain gate may be used in a quantum computation (3]..