This is an updated version of supplementary information to accompany "Quantum supremacy using a programmable superconducting processor", an article published in the October 24, 2019 issue of Nature. The main article is freely available at this https URL. Summary of changes since arXiv:1910.11333v1 (submitted 23 Oct 2019): added URL for qFlex source code; added Erratum section; added Figure S41 comparing statistical and total uncertainty for log and linear XEB; new References [1,65]; miscellaneous updates for clarity and style consistency; miscellaneous typographical and formatting corrections.

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Supplementary information for "Quantum supremacy using a programmable superconducting processor"

The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor1. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits2-7 to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 253 (about 1016). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times-our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy8-14 for this specific computational task, heralding a much-anticipated computing paradigm.

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Quantum supremacy using a programmable superconducting processor

Quantum-state engineering, i.e., active control over the coherent dynamics of suitable quantum-mechanical systems, has become a fascinating prospect of modern physics. With concepts developed in atomic and molecular physics and in the context of NMR, the field has been stimulated further by the perspectives of quantum computation and communication. Low-capacitance Josephson tunneling junctions offer a promising way to realize quantum bits (qubits) for quantum information processing. The article reviews the properties of these devices and the practical and fundamental obstacles to their use. Two kinds of device have been proposed, based on either charge or phase (flux) degrees of freedom. Single- and two-qubit quantum manipulations can be controlled by gate voltages in one case and by magnetic fields in the other case. Both kinds of device can be fabricated with present technology. In flux qubit devices, an important milestone, the observation of superpositions of different flux states in the system eigenstates, has been achieved. The Josephson charge qubit has even demonstrated coherent superpositions of states readable in the time domain. There are two major problems that must be solved before these devices can be used for quantum information processing. One must have a long phase coherence time, which requires that external sources of dephasing be minimized. The review discusses relevant parameters and provides estimates of the decoherence time. Another problem is in the readout of the final state of the system. This issue is illustrated with a possible realization by a single-electron transistor capacitively coupled to the Josephson device, but general properties of measuring devices are also discussed. Finally, the review describes how the basic physical manipulations on an ideal device can be combined to perform useful operations.

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Quantum-state engineering with Josephson-junction devices

5-7 as a candidate for a quantum bit or 'qubit'—the basic component of a quantum computer. Here we report the observation of quantum oscillations in a single- Cooper-pair box. By applying a short voltage pulse via a gate electrode, we can control the coherent quantum state evolution: the pulse modifies the energies of the two charge states non- adiabatically, bringing them into resonance. The resulting state— a superposition of the two charge states—is detected by a tunnelling current through a probe junction. Our results demon- strate electrical coherent control of a qubit in a solid-state

Coherent control of macroscopic quantum states in a single-Cooper-pair box

Superconducting circuits are macroscopic in size but have generic quantum properties such as quantized energy levels, superposition of states, and entanglement, all of which are more commonly associated with atoms. Superconducting quantum bits (qubits) form the key component of these circuits. Their quantum state is manipulated by using electromagnetic pulses to control the magnetic flux, the electric charge or the phase difference across a Josephson junction (a device with nonlinear inductance and no energy dissipation). As such, superconducting qubits are not only of considerable fundamental interest but also might ultimately form the primitive building blocks of quantum computers.

Superconducting quantum bits

Previously we have presented evidence for stripe order of holes and spins in ${\mathrm{La}}_{1.6\ensuremath{-}x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ with $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.12$. Here we show, via neutron diffraction measurements of magnetic scattering, that similar order occurs in crystals with $x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.15$ and $0.20$. Zero-field-cooled magnetization measurements show that all three compositions are also superconducting, with the superconducting transition temperature increasing as the low-temperature staggered magnetization decreases. These results directly demonstrate an intimate connection between stripe correlations and superconductivity.

https://link.aps.org/pdf/10.1103/PhysRevLett.78.338

Coexistence of, and Competition between, Superconductivity and Charge-Stripe Order in La 1 . 6 − x Nd 0.4 Sr x CuO 4

We present a neutron diffraction study of charge and spin order within the ${\mathrm{CuO}}_{2}$ planes of ${\mathrm{La}}_{1.48}$${\mathrm{Nd}}_{0.4}$${\mathrm{Sr}}_{0.12}$${\mathrm{CuO}}_{4}$, a crystal in which superconductivity is anomalously suppressed. At low temperatures we observe elastic magnetic superlattice peaks of the type (1/2\ifmmode\pm\else\textpm\fi{}\ensuremath{\epsilon},1/2,0) and charge-order peaks at (2\ifmmode\pm\else\textpm\fi{}2\ensuremath{\epsilon},0,0), where \ensuremath{\epsilon}=0.118. After cooling the crystal through the low-temperature-orthorhombic (LTO) to low-temperature-tetragonal (LTT) phase transition near 70 K, the charge-order peaks appear first at \ensuremath{\sim}60 K, with the magnetic peaks appearing below 50 K. The magnetic peaks increase in intensity by an order of magnitude below 3 K due to ordering of the Nd ions. We show that the observed diffraction features are consistent with stripe-phase order, in which the dopant-induced holes collect in domain walls that separate antiferromagnetic antiphase domains. The Q dependence of the magnetic scattering indicates that the low-temperature correlation length within the planes is substantial (\ensuremath{\sim}170 \AA{}), but only very weak correlations exist between next-nearest-neighbor planes. Correlations between nearest-neighbor layers are frustrated by pinning of the charge stripes to the lattice distortions of the LTT phase. The spin-density-wave amplitude corresponds to a Cu moment of 0.10\ifmmode\pm\else\textpm\fi{}0.03 ${\mathrm{\ensuremath{\mu}}}_{\mathit{B}}$. The behavior of the electrical resistivity within the LTT phase is examined, and the significance of stripe-phase correlations for understanding the unusual transport properties of layered cuprates is discussed. \textcopyright{} 1996 The American Physical Society.

Neutron-scattering study of stripe-phase order of holes and spins in La1.48Nd0.4Sr0.12CuO4.

Infrared reflectivity measurements on high-quality single crystals of ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ were performed with polarization perpendicular to the ${\mathrm{CuO}}_{2}$ planes (E\ensuremath{\parallel}c). The c-axis spectrum shows a dramatic change at ${\mathit{T}}_{\mathit{c}}$, from a featureless spectrum above ${\mathit{T}}_{\mathit{c}}$ to the one with a sharp reflectivity edge below ${\mathit{T}}_{\mathit{c}}$. We find that the spectrum below ${\mathit{T}}_{\mathit{c}}$ is dominated by the plasma edge arising from the carriers condensed in the superconducting state, not by a superconductong gap excitation, and that the superconducting transition is accompanied by the onset of coherent charge transport along the c axis.

Charge dynamics across the CuO2 planes in La2-xSrxCuO4.

We study Cooper-pair tunneling in a voltage-biased superconducting single-electron transistor under microwave irradiation. By tracing the peak positions of a photon-assisted Josephson-quasiparticle current as a function of the microwave frequency, we observe an energy-dispersion curve in the quasicharge space. This shows that energy-level splitting occurs between two macroscopic quantum states of charge coherently superposed by Josephson coupling. {copyright} {ital 1997} {ital The American Physical Society}

Spectroscopy of Energy-Level Splitting between Two Macroscopic Quantum States of Charge Coherently Superposed by Josephson Coupling

The anisotropic normal-state transport properties, resistivity and thermopower, are investigated for high-quality single crystals of ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$ in the compositional range 0.10\ensuremath{\le}x\ensuremath{\le}0.30. The results indicate that charge transport in the material with superconducting compositions is two-dimensional. A crossover from two to three dimensions is observed at high temperature or on entering the nonsuperconducting overdoped region.

Yoshimichi Nakamura

Papers

Coexistence of, and Competition between, Superconductivity and Charge-Stripe Order in La 1 . 6 − x Nd 0.4 Sr x CuO 4

Neutron-scattering study of stripe-phase order of holes and spins in La1.48Nd0.4Sr0.12CuO4.

Charge dynamics across the CuO2 planes in La2-xSrxCuO4.

Spectroscopy of Energy-Level Splitting between Two Macroscopic Quantum States of Charge Coherently Superposed by Josephson Coupling

Anisotropic transport properties of single-crystal La 2 − x Sr x CuO 4 : Evidence for the dimensional crossover