Efficient Z gates for quantum computing

Abstract

For superconducting qubits, microwave pulses drive rotations around the Bloch sphere. The phase of these drives can be used to generate zero-duration arbitrary "virtual" Z-gates which, combined with two X π/2 gates, can generate any SU(2) gate. Here we show how to best utilize these virtual Z-gates to both improve algorithms and correct pulse errors. We perform randomized benchmarking using a Clifford set of Hadamard and Z-gates and show that the error per Clifford is reduced versus a set consisting of standard finite-duration X and Y gates. Z-gates can correct unitary rotation errors for weakly anharmonic qubits as an alternative to pulse shaping techniques such as DRAG. We investigate leakage and show that a combination of DRAG pulse shaping to minimize leakage and Z-gates to correct rotation errors (DRAGZ) realizes a 13.3 ns X π/2 gate characterized by low error (1.95[3] × 10 −4 ) and low leakage (3.1[6] × 10 −6 ). Ultimately leakage is limited by the finite temperature of the qubit, but this limit is two orders-of-magnitude smaller than pulse errors due to decoherence.Computers based on quantum bits (qubits) are predicted to outperform classical computers for certain critical problems, e.g., factoring. Unlike a classical bit, which is discretely in the state 0 or 1, a qubit can be in a superposition state |Ψ = cos(θ/2)|0 + e iφ sin(θ/2)|1 where |0 and |1 are the quantum versions of the classical 0 and 1 states. This single-qubit superposition state can be geometrically represented as a point on the surface of a unit-sphere known as the Bloch sphere. Critical to implementing a quantum computer is the ability to control the state of the qubit, i.e., transform the qubit state arbitrarily between two points on the Bloch sphere. This is accomplished by unitary transformations (gates), which correspond to rotations of the state around different axes in the Bloch sphere representation. Physically, X and Y gates (rotations around the X and Y axes) are generated by modulating the coupling between the states |0 and |1 at the frequency difference between these states ω 01 = (E |1 −E |0 )/h. This modulation drive has the general form Ω(t) cos(ω D t − γ) where Ω(t) is the drive strength of the rotation, ω D is the drive frequency (ω D = ω 01 on resonance) and γ is the drive phase. The duration of the gate is set by the desired rotation angle and the drive strength. On-resonance, when γ = 0, the qubit state rotates around the X axis and when γ = π 2 the rotation is around the Y axis. Therefore, the geometric X and Y axes in the Bloch sphere correspond to a real π 2 phase difference between drive fields. Rotations around the remaining axis (Z axis), i.e., Zgates, correspond to a change in the relative phase between the |0 and |1 states. A Z-gate can be implemented by either detuning the frequency of the qubit with respect to the drive field for some finite amount of time (e.g. see Ref.[1]) or by composite X and Y gates. The result is that the qubit state rotates with respect to the X and Y axes. However, it is equivalent t...

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“…The qubits are controlled by the external control field n gi (t), which is directly proportional to the voltage pulses applied to the qubit. Thus quantum gates are implemented by choosing a certain pulse form for n gi (t) (see [22,23]). The resonator is described by raising and lowering operatorsâ † andâ, respectively.…”

“…(2). We consider a generic sum of shaped microwave pulses applied on each qubit as described in [23], namely…”

“…We utilize the virtual Z gate (VZ gate) described in [23] and used in the IBMQX [5] to implement rotations about the z axis. This means that instead of applying another pulse, we simply change the phase γ of all the following pulses (see Appendix B for details).…”

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