skip to main content

Title: Random Quantum Circuits Transform Local Noise into Global White Noise

We study the distribution over measurement outcomes of noisy random quantum circuits in the regime of low fidelity, which corresponds to the setting where the computation experiences at least one gate-level error with probability close to one. We model noise by adding a pair of weak, unital, single-qubit noise channels after each two-qubit gate, and we show that for typical random circuit instances, correlations between the noisy output distribution$$p_{\text {noisy}}$$pnoisyand the corresponding noiseless output distribution$$p_{\text {ideal}}$$pidealshrink exponentially with the expected number of gate-level errors. Specifically, the linear cross-entropy benchmarkFthat measures this correlation behaves as$$F=\text {exp}(-2s\epsilon \pm O(s\epsilon ^2))$$F=exp(-2sϵ±O(sϵ2)), where$$\epsilon $$ϵis the probability of error per circuit location andsis the number of two-qubit gates. Furthermore, if the noise is incoherent—for example, depolarizing or dephasing noise—the total variation distance between the noisy output distribution$$p_{\text {noisy}}$$pnoisyand the uniform distribution$$p_{\text {unif}}$$punifdecays at precisely the same rate. Consequently, the noisy output distribution can be approximated as$$p_{\text {noisy}}\approx Fp_{\text {ideal}}+ (1-F)p_{\text {unif}}$$pnoisyFpideal+(1-F)punif. In other words, although at least one local error occurs with probability$$1-F$$1-F, the errors are scrambled by the random quantum circuit and can be treated as global white noise, contributing completely uniform output. Importantly, we upper bound the average total variation error in this approximation by$$O(F\epsilon \sqrt{s})$$O(Fϵs). Thus, the “white-noise approximation” is meaningful when$$\epsilon \sqrt{s} \ll 1$$ϵs1, a quadratically weaker condition than the$$\epsilon s\ll 1$$ϵs1requirement to maintain high fidelity. The bound applies if the circuit size satisfies$$s \ge \Omega (n\log (n))$$sΩ(nlog(n)), which corresponds to onlylogarithmic depthcircuits, and if, additionally, the inverse error rate satisfies$$\epsilon ^{-1} \ge {\tilde{\Omega }}(n)$$ϵ-1Ω~(n), which is needed to ensure errors are scrambled faster thanFdecays. The white-noise approximation is useful for salvaging the signal from a noisy quantum computation; for example, it was an underlying assumption in complexity-theoretic arguments that noisy random quantum circuits cannot be efficiently sampled classically, even when the fidelity is low. Our method is based on a map from second-moment quantities in random quantum circuits to expectation values of certain stochastic processes for which we compute upper and lower bounds.

more » « less
Author(s) / Creator(s):
; ;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Communications in Mathematical Physics
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    A search is reported for pairs of light Higgs bosons ($${\textrm{H}} _1$$H1) produced in supersymmetric cascade decays in final states with small missing transverse momentum. A data set of LHC$$\hbox {pp}$$ppcollisions collected with the CMS detector at$$\sqrt{s}=13\,\text {TeV} $$s=13TeVand corresponding to an integrated luminosity of 138$$\,\text {fb}^{-1}$$fb-1is used. The search targets events where both$${\textrm{H}} _1$$H1bosons decay into Equation missing<#comment/>pairs that are reconstructed as large-radius jets using substructure techniques. No evidence is found for an excess of events beyond the background expectations of the standard model (SM). Results from the search are interpreted in the next-to-minimal supersymmetric extension of the SM, where a “singlino” of small mass leads to squark and gluino cascade decays that can predominantly end in a highly Lorentz-boosted singlet-like$${\textrm{H}} _1$$H1and a singlino-like neutralino of small transverse momentum. Upper limits are set on the product of the squark or gluino pair production cross section and the square of the Equation missing<#comment/>branching fraction of the$${\textrm{H}} _1$$H1in a benchmark model containing almost mass-degenerate gluinos and light-flavour squarks. Under the assumption of an SM-like Equation missing<#comment/>branching fraction,$${\textrm{H}} _1$$H1bosons with masses in the range 40–120$$\,\text {GeV}$$GeVarising from the decays of squarks or gluinos with a mass of 1200–2500$$\,\text {GeV}$$GeVare excluded at 95% confidence level.

    more » « less
  2. Abstract

    We present a proof of concept for a spectrally selective thermal mid-IR source based on nanopatterned graphene (NPG) with a typical mobility of CVD-grown graphene (up to 3000$$\hbox {cm}^2\,\hbox {V}^{-1}\,\hbox {s}^{-1}$$cm2V-1s-1), ensuring scalability to large areas. For that, we solve the electrostatic problem of a conducting hyperboloid with an elliptical wormhole in the presence of anin-planeelectric field. The localized surface plasmons (LSPs) on the NPG sheet, partially hybridized with graphene phonons and surface phonons of the neighboring materials, allow for the control and tuning of the thermal emission spectrum in the wavelength regime from$$\lambda =3$$λ=3to 12$$\upmu$$μm by adjusting the size of and distance between the circular holes in a hexagonal or square lattice structure. Most importantly, the LSPs along with an optical cavity increase the emittance of graphene from about 2.3% for pristine graphene to 80% for NPG, thereby outperforming state-of-the-art pristine graphene light sources operating in the near-infrared by at least a factor of 100. According to our COMSOL calculations, a maximum emission power per area of$$11\times 10^3$$11×103W/$$\hbox {m}^2$$m2at$$T=2000$$T=2000K for a bias voltage of$$V=23$$V=23V is achieved by controlling the temperature of the hot electrons through the Joule heating. By generalizing Planck’s theory to any grey body and deriving the completely general nonlocal fluctuation-dissipation theorem with nonlocal response of surface plasmons in the random phase approximation, we show that the coherence length of the graphene plasmons and the thermally emitted photons can be as large as 13$$\upmu$$μm and 150$$\upmu$$μm, respectively, providing the opportunity to create phased arrays made of nanoantennas represented by the holes in NPG. The spatial phase variation of the coherence allows for beamsteering of the thermal emission in the range between$$12^\circ$$12and$$80^\circ$$80by tuning the Fermi energy between$$E_F=1.0$$EF=1.0eV and$$E_F=0.25$$EF=0.25eV through the gate voltage. Our analysis of the nonlocal hydrodynamic response leads to the conjecture that the diffusion length and viscosity in graphene are frequency-dependent. Using finite-difference time domain calculations, coupled mode theory, and RPA, we develop the model of a mid-IR light source based on NPG, which will pave the way to graphene-based optical mid-IR communication, mid-IR color displays, mid-IR spectroscopy, and virus detection.

    more » « less
  3. Abstract

    The double differential cross sections of the Drell–Yan lepton pair ($$\ell ^+\ell ^-$$+-, dielectron or dimuon) production are measured as functions of the invariant mass$$m_{\ell \ell }$$m, transverse momentum$$p_{\textrm{T}} (\ell \ell )$$pT(), and$$\varphi ^{*}_{\eta }$$φη. The$$\varphi ^{*}_{\eta }$$φηobservable, derived from angular measurements of the leptons and highly correlated with$$p_{\textrm{T}} (\ell \ell )$$pT(), is used to probe the low-$$p_{\textrm{T}} (\ell \ell )$$pT()region in a complementary way. Dilepton masses up to 1$$\,\text {Te\hspace{-.08em}V}$$TeVare investigated. Additionally, a measurement is performed requiring at least one jet in the final state. To benefit from partial cancellation of the systematic uncertainty, the ratios of the differential cross sections for various$$m_{\ell \ell }$$mranges to those in the Z mass peak interval are presented. The collected data correspond to an integrated luminosity of 36.3$$\,\text {fb}^{-1}$$fb-1of proton–proton collisions recorded with the CMS detector at the LHC at a centre-of-mass energy of 13$$\,\text {Te\hspace{-.08em}V}$$TeV. Measurements are compared with predictions based on perturbative quantum chromodynamics, including soft-gluon resummation.

    more » « less
  4. Abstract

    The azimuthal ($$\Delta \varphi $$Δφ) correlation distributions between heavy-flavor decay electrons and associated charged particles are measured in pp and p–Pb collisions at$$\sqrt{s_{\mathrm{{NN}}}} = 5.02$$sNN=5.02TeV. Results are reported for electrons with transverse momentum$$44<pT<16$$\textrm{GeV}/c$$GeV/c and pseudorapidity$$|\eta |<0.6$$|η|<0.6. The associated charged particles are selected with transverse momentum$$11<pT<7$$\textrm{GeV}/c$$GeV/c, and relative pseudorapidity separation with the leading electron$$|\Delta \eta | < 1$$|Δη|<1. The correlation measurements are performed to study and characterize the fragmentation and hadronization of heavy quarks. The correlation structures are fitted with a constant and two von Mises functions to obtain the baseline and the near- and away-side peaks, respectively. The results from p–Pb collisions are compared with those from pp collisions to study the effects of cold nuclear matter. In the measured trigger electron and associated particle kinematic regions, the two collision systems give consistent results. The$$\Delta \varphi $$Δφdistribution and the peak observables in pp and p–Pb collisions are compared with calculations from various Monte Carlo event generators.

    more » « less
  5. Abstract

    We prove that$${{\,\textrm{poly}\,}}(t) \cdot n^{1/D}$$poly(t)·n1/D-depth local random quantum circuits with two qudit nearest-neighbor gates on aD-dimensional lattice withnqudits are approximatet-designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was$${{\,\textrm{poly}\,}}(t)\cdot n$$poly(t)·ndue to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for$$D=1$$D=1. We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy ($${{\,\mathrm{\textsf{PH}}\,}}$$PH) is infinite and that certain counting problems are$$\#{\textsf{P}}$$#P-hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that$$O(\sqrt{n})$$O(n)depth suffices for anti-concentration. The proof is based on a previous construction oft-designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size$$O(n\ln ^2 n)$$O(nln2n)corresponding to depth$$O(\ln ^3 n)$$O(ln3n). We also show a lower bound of$$\Omega (n \ln n)$$Ω(nlnn)for the size of such circuit in this case. We also prove that anti-concentration is possible in depth$$O(\ln n \ln \ln n)$$O(lnnlnlnn)(size$$O(n \ln n \ln \ln n)$$O(nlnnlnlnn)) using a different model.

    more » « less