Aba: Youngseok Kim Andrew Eddins Sajant Anand Ken Xuan Wei Ewout van den Berg Sami Rosenblatt Hasan Nayfeh Yantao Wu Michael Zaletel Kristan Temme Abhinav Kandala Abstract ICompute yawo kupeleka liwiro likulu k Bernard classical counterpart kwa mavutha ena. Komabe, chopinga chachikulu kwambiri pofuna kukwaniritsa mphamvu zake zonse ndi phokoso lomwe liripo m'malo awa. Yankho lomwe lakhala likuvomerezedwa ndi okha ndi kukhazikitsa ma circuits a quantum omwe alibe zolakwika, omwe sakukwanira kwa owonetsa apano. Pano tikunena za kuyesera pa processor ya noisy ya 127-qubit ndipo tikuwonetsa kuyeza kwa magawo olondola a voliyumu ya dera pamlingo wopitilira masamu apakompyuta a brute-force. Tikunena kuti izi zikuwonetsa umboni wa ntchito ya quantum computing m'nthawi ya pre-fault-tolerant. Zotsatira zoyeserazo zimayendetsedwa ndi kupita patsogolo kwa kukhazikika ndi kukhazikitsa kwa superconducting processor pamlingo uwu komanso kutha kulemba mawonekedwe ndi kuyendetsa phokoso molamulidwa pa chipangizo chachikulu chonchi. Tikuwonetsa kulondola kwa magawo oyesedwa poyerekeza ndi zotsatira za ma circuits omwe angatsimikizidwe molondola. M'malo olamulira omwe ali ndi mphamvu zambiri, kompyuta ya quantum imapereka zotsatira zolondola zomwe njira zapamwamba kwambiri zamakono ngati 1D yochokera pa pure-state (matrix product states, MPS) ndi 2D (isometric tensor network states, isoTNS) sizigwira ntchito. Kuyeseraku kumawonetsa chida champhamvu chofunikira pakukwaniritsa ntchito zapafupi za quantum. Main Ndizovomerezeka kuti njira zapamwamba za quantum algorithms monga kupanga kapena kuwerengera gawo zidzafunikira kukonza zolakwika za quantum. Komabe, ndizokambitsana mwamphamvu ngati owonetsa omwe alipo pano angapangidwe kukhala odalirika okwanira kuti azitha kuyendetsa ma circuits ena achidule achidule pamlingo womwe ungabweretse mwayi wazovuta zamasiku ano. Pakadali pano, kuyembekezera kwachizolowezi ndikuti kukhazikitsa ngakhale ma circuits a quantum osavuta omwe ali ndi mwayi wopitilira luso lamakompyuta adzayenera kudikirira mpaka owonetsa apamwamba, olola zolakwika atafika. Ngakhale kupita patsogolo kwakukulu kwa hardware ya quantum m'zaka zaposachedwa, malire osavuta achilungamo amathandizira ulosi wosasangalatsa uwu; munthu akuwerengera kuti dera la quantum la 100 qubits lalikulu ndi 100 gate-layers ozama lopangidwa ndi 0.1% gate error limatulutsa chiyero cha state chocheperapo 5 × 10−4. Komabe, funso likukhalabe ngati zinthu za state yabwino zitha kupezeka ngakhale ndi zolakwika zochepa chonchi. Njira yochotsera cholakwika yamwayi wa quantum pamasamba ovuta imayankha funso ili mwachindunji, ndiye kuti, kuti munthu amatha kupanga magawo olondola kuchokera pamaulendo osiyanasiyana a quantum circuit ovuta pogwiritsa ntchito classical post-processing. Mwayi wa quantum ukhoza kuyandikira m'njira ziwiri: choyamba, powonetsera kuthekera kwa zipangizo zomwe zilipo kuti zigwire ntchito yolondola pamlingo womwe uli kupitilira masamu a brute-force, ndipo pambuyo pake kupeza mavuto okhala ndi ma circuits a quantum omwe amalandira mwayi kuchokera kuzipangizo izi. Pano tikuyang'ana pakuchita sitepe yoyamba ndipo sitikufuna kukhazikitsa ma circuits a quantum pazovuta zomwe zalola liwiro. Timagwiritsa ntchito superconducting quantum processor yokhala ndi qubits 127 kuti tiyendetse ma circuits a quantum okhala ndi masiteji okwana 60 a two-qubit gates, okwana 2,880 CNOT gates. Ma circuit apadera amtunduwu nthawi zambiri amakhala ovuta pama njira za brute-force zapakompyuta. Chifukwa chake timayang'ana poyamba pa milandu yapadera ya ma circuits omwe amalola kutsimikizika kwapakompyuta kwa magawo oyesedwa. Kenako timatembenukira kumadera a dera ndi owonera omwe masamu akukhala ovuta ndipo timayerekeza ndi zotsatira zochokera pama njira zapamwamba kwambiri zamakono. Dera lathu lopangira dera ndi Trotterized nthawi yoyenda ya 2D transverse-field Ising model, kugawana topology ya processor ya qubit (Fig.a). The Ising model appears extensively across several areas in physics and has found creative extensions in recent simulations exploring quantum many-body phenomena, such as time crystals, quantum scars and Majorana edge modes. Ngati kuyesa kwa ntchito ya quantum computing, komabe, nthawi yoyenda ya 2D transverse-field Ising model ndiyofunikira kwambiri pamlingo wa kukula kwa mphamvu komwe njira zapamwamba zamakono zimavutika. , Munthu aliyense wa Trotter wa Ising simulation amaphatikizapo single-qubit ndi two-qubit rotations. Random Pauli gates are inserted to twirl (spirals) and controllably scale the noise of each CNOT layer. The dagger indicates conjugation by the ideal layer. , Three depth-1 layers of CNOT gates suffice to realize interactions between all neighbour pairs on ibm_kyiv. , Characterization experiments efficiently learn the local Pauli error rates , (colour scales) comprising the overall Pauli channel Λ associated with the th twirled CNOT layer. (Figure expanded in Supplementary Information IV.A). , Pauli errors inserted at proportional rates can be used to either cancel (PEC) or amplify (ZNE) the intrinsic noise. a X ZZ b c λl i l l d Makamaka, timaganizira za nthawi yothamanga ya Hamiltonian, kumene > 0 ndi kulumikizana kwa spin wapafupi kwambiri ndi < ndi ndi gawo lalikulu la transverse. Spin dynamics from an initial state can be simulated by means of first-order Trotter decomposition of the time-evolution operator, J i j h kumene nthawi yoyendayenda imadulidwa mu / Trotter steps and ndi are ndi rotation gates, motsatira. Sitikukhudzidwa ndi cholakwika cha mtundu chifukwa cha Trotterization ndipo chifukwa chake timatenga dera loterolo ngati labwino pama comparative onse. Kuti muwoneke mosavuta, timayang'ana pa nkhani = −2 = −π/2 kotero kuti rotation imafuna CNOT imodzi yokha, T T δt ZZ X θJ Jδt ZZ kumene equality holds up to a global phase. Mu dera lotsatira (Fig.a), munthu aliyense wa Trotter amagwirizana ndi gawo la single-qubit rotations, R ( ), kutsatiridwa ndi magawo ogwirizana a parallelized two-qubit rotations, R ( ). X θh ZZ θJ Kwa kukhazikitsidwa kwachidziwitso, tidagwiritsa ntchito IBM Eagle processor ibm_kyiv, yopangidwa ndi ma qubits 127 okhazikika a transmon okhala ndi heavy-hex connectivity ndi median 1 ndi 2 nthawi za 288 μs ndi 127 μs, motsatira. Nthawi izi zokhazikika ndi zazikulu kwambiri pazigawo zazikulu za superconducting ndipo zimalola kuya kwa dera lomwe likuchitika m'nthawi yathu. Ma CNOT gates awiri pakati pa oyandikana nawo amachitika podutsa cross-resonance interaction. Popeza qubit iliyonse ili ndi oyandikana nawo osachepera atatu, zonse interactions zitha kuchitika m'magawo atatu a parallelized CNOT gates (Fig.b). Ma CNOT gates mkati mwa gawo lililonse amayesedwa kuti agwire ntchito limodzi moyenera (onani Methods kuti mudziwe zambiri). T T ZZ Tsopano tikuwona kuti kusintha kwa magwiridwe antchito kwa hardware kumeneku kumayambitsa mavuto akuluakulu kuti achite bwino ndi cholakwika, poyerekeza ndi ntchito zaposachedwa pamalo awa. Probabilistic error cancellation (PEC) yasonyezedwa kukhala yothandiza kwambiri popereka ziwerengero zosalowerera za owonera. Mu PEC, mtundu wodziwika wa phokoso umaphunziridwa ndipo umasinthidwa mwa kuyimira kuchokera pogaŵiridwa kwa ma circuits ovuta omwe amagwirizana ndi mtunduwo. Komabe, pakadali pano pali zolakwika pazida zathu, chiwopalawanso cha kuyimitsidwa kwa ma voliyumu a dera omwe ali m'nthawi yathu ikupitilirabe, monga momwe takambirana pansipa. Chifukwa chake timatembenukira ku zero-noise extrapolation (ZNE), yomwe imapereka chiwerengero chosayenera pamitengo yotsika kwambiri yoyimitsidwa. ZNE ndi njira yofananira kapena njira yochotsa yamagawo osakhazikika ngati ntchito ya cholakwika. Izi zimafuna kusinthasintha kwa phokoso la hardware la chipangizo ndi nambala yodziwika ya kupeza kuti ifotokoze malire a = 0. ZNE yakhala ikugwiritsidwa ntchito kwambiri chifukwa njira zowonjezeretsira phokoso zochokera pamasinthidwe opangira pulse kapena kubwereza kwa subcircuit zapangitsa kuti pakhale kufunikira koyenera kwa phokoso, pomwe tikudalira malingaliro osavuta pamavuto a chipangizocho. Komabe, kulimbikitsidwa kwa phokoso koyenera kumatha kubweretsa kuchepa kwakukulu kwa kusayenera kwa chiwerengero chochotsedwa, monga momwe tikuwonetsera pano. G G The sparse Pauli–Lindblad noise model proposed in ref. turns out to be especially well suited for noise shaping in ZNE. The model takes the form , in which is a Lindbladian comprising Pauli jump operators weighted by rates . It was shown in ref. that restricting to jump operators acting on local pairs of qubits yields a sparse noise model that can be efficiently learned for many qubits and that accurately captures the noise associated with layers of two-qubit Clifford gates, including crosstalk, when combined with random Pauli twirls. The noisy layer of gates is modelled as a set of ideal gates preceded by some noise channel Λ. Thus, applying Λ before the noisy layer produces an overall noise channel Λ with gain = + 1. Given the exponential form of the Pauli–Lindblad noise model, the map is obtained by simply multiplying the Pauli rates by . The resulting Pauli map can be sampled to obtain appropriate circuit instances; for ≥ 0, the map is a Pauli channel that can be sampled directly, whereas for < 0, quasi-probabilistic sampling is needed with sampling overhead −2 for some model-specific . In PEC, we choose = −1 to obtain an overall zero-gain noise level. In ZNE, we instead amplify the noise to different gain levels and estimate the zero-noise limit using extrapolation. For practical applications, we need to consider the stability of the learned noise model over time (Supplementary Information III.A), for instance, owing to qubit interactions with fluctuating microscopic defects known as two-level systems. Pi λi α G G α λi α α α γ α γ α Clifford circuits serve as useful benchmarks of estimates produced by error mitigation, as they can be efficiently simulated classically. Notably, the entire Ising Trotter circuit becomes Clifford when is chosen to be a multiple of π/2. As a first example, we therefore set the transverse field to zero (R (0) = ) and evolve the initial state |0⟩⊗127 (Fig.a). The CNOT gates nominally leave this state unchanged, so the ideal weight-1 observables all have expectation value 1; owing to the Pauli twirling of each layer, the bare CNOTs do affect the state. For each Trotter experiment, we first characterized the noise models Λ for the three Pauli-twirled CNOT layers (Fig.c) and then used these models to implement Trotter circuits with noise gain levels ∈ {1, 1.2, 1.6}. Figurea illustrates the estimation of ⟨ 106⟩ after four Trotter steps (12 CNOT layers). For each , we generated 2,000 circuit instances in which, before each layer , we have inserted products of one-qubit and two-qubit Pauli errors from drawn with probabilities and executed each instance 64 times, totalling 384,000 executions. As more circuit instances are accumulated, the estimates of ⟨ 106⟩ , corresponding to the different gains , converge to distinct values. The different estimates are then fit by an extrapolating function in to estimate the ideal value ⟨ 106⟩0. The results in Fig.a highlight the reduced bias from exponential extrapolation in comparison with linear extrapolation. That said, exponential extrapolation can exhibit instabilities, for instance, when expectation values are unresolvably close to zero, and—in such cases—we iteratively downgrade the extrapolation model complexity (see Supplementary Information II.B). The procedure outlined in Fig.a was applied to the measurement results from each qubit to estimate all = 127 Pauli expectations ⟨ ⟩0. The variation in the unmitigated and mitigated observables in Fig.b is indicative of the non-uniformity in the error rates across the entire processor. We report the global magnetization along , , for increasing depth in Fig.c. Although the unmitigated result shows a gradual decay from 1 with an increasing deviation for deeper circuits, ZNE greatly improves agreement, albeit with a small bias, with the ideal value even out to 20 Trotter steps, or 60 CNOT depth. Notably, the number of samples used here is much smaller than an estimate of the sampling overhead that would be needed in a naive PEC implementation (see Supplementary Information IV.B). In principle, this disparity may be greatly reduced by more advanced PEC implementations using light-cone tracing or by improvements in hardware error rates. As future hardware and software developments bring down sampling costs, PEC may be preferred when affordable to avoid the potentially biased nature of ZNE. θh X I Zq l G Z G l i Z G G G Z q N Zq Mitigated expectation values from Trotter circuits at the Clifford condition = 0. , Convergence of unmitigated ( = 1), noise-amplified ( > 1) and noise-mitigated (ZNE) estimates of ⟨ 106⟩ after four Trotter steps. In all panels, error bars indicate 68% confidence intervals obtained by means of percentile bootstrap. Exponential extrapolation (exp, dark blue) tends to outperform linear extrapolation (linear, light blue) when differences between the converged estimates of ⟨ 106⟩ ≠0 are well resolved. , Magnetization (large markers) is computed as the mean of the individual estimates of ⟨ ⟩ for all qubits (small markers). , As circuit depth is increased, unmitigated estimates of decay monotonically from the ideal value of 1. ZNE greatly improves the estimates even after 20 Trotter steps (see Supplementary Information II for ZNE details). θh a G G Z Z G b Zq c Mz Next, we test the efficacy of our methods for non-Clifford circuits and the Clifford = π/2 point, with non-trivial entangling dynamics compared with the identity-equivalent circuits discussed in Fig.. The non-Clifford circuits are of particular importance to test, as the validity of exponential extrapolation is no longer guaranteed (see Supplementary Information V and ref.). We restrict the circuit depth to five Trotter steps (15 CNOT layers) and judiciously choose observables that are exactly verifiable. Figure shows the results as is swept between 0 and π/2 for three such observables of increasing weight. Figurea shows as before, an average of weight-1 ⟨ ⟩ observables, whereas Fig.b,c show weight-10 and weight-17 observables. The latter operators are stabilizers of the Clifford circuit at = π/2, obtained by evolution of the initial stabilizers 13 and 58, respectively, of |0⟩⊗127 for five Trotter steps, ensuring non-vanishing expectation values in the strongly entangling regime of particular interest. Although the entire 127-qubit circuit is executed experimentally, light-cone and depth-reduced (LCDR) circuits enable brute-force classical simulation of the magnetization and weight-10 operator at this depth (see Supplementary Information VII). Over the full extent of the sweep, the error-mitigated observables show good agreement with the exact evolution (see Fig.a,b). However, for the weight-17 operator, the light cone expands to 68 qubits, a scale beyond brute-force classical simulation, so we turn to tensor network methods. θh θh Mz Z θh Z Z θh Expectation value estimates for sweeps at a fixed depth of five Trotter steps for the circuit in Fig.a. The considered circuits are non-Clifford except at = 0, π/2. Light-cone and depth reductions of respective circuits enable exact classical simulation of the observables for all . For all three plotted quantities (panel titles), mitigated experimental results (blue) closely track the exact behaviour (grey). In all panels, error bars indicate 68% confidence intervals obtained by means of percentile bootstrap. The weight-10 and weight-17 observables in and are stabilizers of the circuit at = π/2 with respective eigenvalues +1 and −1; all values in have been negated for visual simplicity. The lower inset in depicts variation of ⟨ ⟩ at = 0.2 across the device before and after mitigation and compares with exact results. Upper insets in all panels illustrate causal light cones, indicating in blue the final qubits measured (top) and the nominal set of initial qubits that can influence the state of the final qubits (bottom). also depends on 126 other cones besides the example shown. Although in all panels exact results are obtained from simulations of only causal qubits, we include tensor network simulations of all 127 qubits (MPS, isoTNS) to help gauge the domain of validity for those techniques, as discussed in the main text. isoTNS results for the weight-17 operator in are not accessible with current methods (see Supplementary Information VI). All experiments were carried out for = 1, 1.2, 1.6 and extrapolated as in Supplementary Information II.B. For each , we generated 1,800–2,000 random circuit instances for and and 2,500–3,000 instances for . θh θh θh b c θh c a Zq θh Mz c G G a b c Tensor networks have been widely used to approximate and compress quantum state vectors that arise in the study of the low-energy eigenstates of and time evolution by local Hamiltonians and, more recently, have been successfully used to simulate low-depth noisy quantum circuits. Simulation accuracy can be improved by increasing the bond dimension , which constrains the amount of entanglement of the represented quantum state, at a computational cost scaling polynomially with . As entanglement (bond dimension) of a generic state grows linearly (exponentially) with time evolution until it saturates the volume law, deep quantum circuits are inherently difficult for tensor networks. We consider both quasi-1D matrix product states (MPS) and 2D isometric tensor network states (isoTNS) that have and scaling of time-evolution complexity, respectively. Details of both methods and their strengths are provided in Methods and Supplementary Information VI. Specifically for the case of the weight-17 operator shown in Fig.c, we find that an MPS simulation of the LCDR circuit at = 2,048 is sufficient to obtain the exact evolution (see Supplementary Information VIII). The larger causal cone of the weight-17 observable results in an experimental signal that is weaker compared with that of the weight-10 observable; nevertheless, mitigation still yields good agreement with the exact trace. This comparison suggests that the domain of experimental accuracy could extend beyond the scale of exact classical simulation. χ χ χ We expect that these experiments will eventually extend to circuit volumes and observables in which such light-cone and depth reductions are no longer important. Therefore, we also study the performance of MPS and isoTNS for the full 127-qubit circuit executed in Fig., at respective bond dimensions of = 1,024 and = 12, which are primarily limited by memory requirements. Figure shows that the tensor network methods struggle with increasing , losing both accuracy and continuity near the verifiable Clifford point = π/2. This breakdown can be understood in terms of entanglement properties of the state. The stabilizer state produced by the circuit at = π/2 has an exactly flat bipartite entanglement spectrum, found from a Schmidt decomposition of a 1D ordering of the qubits. Thus, truncating states with small Schmidt weight—the basis of all tensor network algorithms—is not justified. However, as exact tensor network representations generically require bond dimension exponential in circuit depth, truncation is necessary for tractable numerical simulations. χ χ θh θh θh Finally, in Fig., we stretch our experiments to regimes in which the exact solution is not available with the classical methods considered here. The first example (Fig.a) is similar to Fig.c but with a further final layer of single-qubit Pauli rotations that interrupt the circuit-depth reduction that previously enabled exact verification for any (see Supplementary Information VII). At the verifiable Clifford point = π/2, the mitigated results agree again with the ideal value, whereas the = 3,072 MPS simulation of the 68-qubit LCDR circuit markedly fails in the strongly entangling regime of interest. Although = 2,048 was sufficient for exact simulation of the weight-17 operator in Fig.c, an MPS bond dimension of 32,768 would be needed for exact simulation of this modified circuit and operator with = π/2. θh θh χ χ θh Plot markers, confidence intervals and causal light cones appear as defined in Fig.. , Estimates of a weight-17 observable (panel title) after five Trotter steps for several values of . The circuit is similar to that in Fig.c but with further single-qubit rotations at the end. This effectively simulates the time evolution of the spins after Trotter step six by using the same number of two-qubit gates used for Trotter step five. As in Fig.c, the observable is a stabilizer at = π/2 with eigenvalue −1, so we negate the axis for visual simplicity. Optimization of the MPS simulation by including only qubits and gates in the causal light cone enables a higher bond dimension ( = 3,072), but the simulation still fails to approach −1 (+1 in negated axis) at = π/2. , Estimates of the single-site magnetization 〈 62〉 after 20 Trotter steps for several values of . The MPS simulation is light-cone-optimized and performed with bond dimension = 1,024, whereas the isoTNS simulation ( = 12) includes the gates outside the light cone. The experiments were carried out with = 1, 1.3, 1.6 for and = 1, 1.2, 1.6 for , and extrapolated as in Supplementary Information II.B. For each , we generated 2,000–3,200 random circuit instances for and 1,700–2,400 instances for . a θh θh y χ y θh b Z θh χ χ G a G b G a b As a final example, we extend the circuit depth to 20 Trotter steps (60 CNOT layers) and estimate the dependence of a weight-1 observable, ⟨ 62⟩, in Fig.b, in which the causal cone extends over the entire device. Given the non-uniformity of device performance, also seen in the spread of single-site observables in Fig.b, we choose an observable that obtains the expected result ⟨ 62⟩ ≈ 1 at the verifiable = 0 point. Despite the greater depth, the MPS simulations of the LCDR circuit agree well with the experiment θh Z Z θh
