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The computational power of random quantum circuits in arbitrary geometries

Matthew DeCross, Reza Haghshenas, Minzhao Liu, Enrico Rinaldi, Johnnie Gray, Yuri Alexeev, Charles H. Baldwin, John P. Bartolotta, Matthew Bohn, Eli Chertkov, Julia Cline, Jonhas Colina, Davide DelVento, Joan M. Dreiling, Cameron Foltz, John P. Gaebler, Thomas M. Gatterman, Christopher N. Gilbreth, Joshua Giles, Dan Gresh, Alex Hall, Aaron Hankin, Azure Hansen, Nathan Hewitt, Ian Hoffman, Craig Holliman, Ross B. Hutson, Trent Jacobs, Jacob Johansen, Patricia J. Lee, Elliot Lehman, Dominic Lucchetti, Danylo Lykov, Ivaylo S. Madjarov, Brian Mathewson, Karl Mayer, Michael Mills, Pradeep Niroula, Juan M. Pino, Conrad Roman, Michael Schecter, Peter E. Siegfried, Bruce G. Tiemann, Curtis Volin, James Walker, Ruslan Shaydulin, Marco Pistoia, Steven. A. Moses, David Hayes, Brian Neyenhuis, Russell P. Stutz, Michael Foss-Feig

arXiv:2406.02501v3quant-ph

TL;DR

Quantum architectures still face a gap between present capabilities and large-scale error-corrected computing, while RCS also depends on fidelity and connectivity. This paper upgrades H2 for 56-qubit arbitrary-connectivity sampling and reports high-fidelity circuits that appear classically challenging, with qubit number the primary constraint.

  • Problem

    Existing quantum architectures remain far from the requirements for large-scale error-corrected algorithms, while their differing strengths in qubit count, error rate, connectivity, and clock speed complicate comparisons.

    Source: §I · p. 1

  • Method

    The authors upgrade Quantinuum’s H2 trapped-ion computer to 56 qubits with arbitrary connectivity and implement random circuit sampling in random geometries.

    Source: §I · p. 2

  • Results

    H2 circuits deep enough to saturate exact tensor-network contraction cost can be executed without error about 35% of the time, and the authors conclude RCS capability is limited mainly by qubit number.

    Source: §I · p. 2

  • Takeaways & Limitations

    The authors conclude that QCCD-based RCS capability can continue to grow rapidly as qubit number scales up.

    Source: §I · p. 2

Abstract

from arXiv · show

Empirical evidence for a gap between the computational powers of classical and quantum computers has been provided by experiments that sample the output distributions of two-dimensional quantum circuits. Many attempts to close this gap have utilized classical simulations based on tensor network techniques, and their limitations shed light on the improvements to quantum hardware required to frustrate classical simulability. In particular, quantum computers having in excess of $\sim 50$ qubits are primarily vulnerable to classical simulation due to restrictions on their gate fidelity and their connectivity, the latter determining how many gates are required (and therefore how much infidelity is suffered) in generating highly-entangled states. Here, we describe recent hardware upgrades to Quantinuum's H2 quantum computer enabling it to operate on up to $56$ qubits with arbitrary connectivity and $99.843(5)\%$ two-qubit gate fidelity. Utilizing the flexible connectivity of H2, we present data from random circuit sampling in highly connected geometries, doing so at unprecedented fidelities and a scale that appears to be beyond the capabilities of state-of-the-art classical algorithms. The considerable difficulty of classically simulating H2 is likely limited only by qubit number, demonstrating the promise and scalability of the QCCD architecture as continued progress is made towards building larger machines.

I. INTRODUCTION

Random circuit sampling (RCS) offers a way to compare quantum and classical computational power, with connectivity and fidelity shaping how difficult circuits are to simulate classically. The paper describes H2 upgrades and uses its arbitrary connectivity to run classically challenging circuits at high fidelity.

I. INTRODUCTION

  • Quantum architectures trade off qubit count, physical error rates, connectivity, and clock speed, with no architecture uniformly superior across these properties.High connectivity may reduce QEC qubit requirements, while slower clock speed may be offset by higher fidelity.Source: §I · p. 1
  • RCS compares quantum and classical computational power by asking computers to sample from a random circuit’s output distribution.The relative run times depend on quantum-computer speed, size, gate fidelity, and connectivity.Source: §I · p. 1
  • Tensor-network methods can simulate imperfect circuits, but higher connectivity generally lowers achievable simulation fidelity at fixed classical resources.Increasing connectivity or fidelity can therefore frustrate these simulation methods.Source: §I · pp. 1–2
  • H2 upgrades enable up to 56 qubits with arbitrary connectivity, and the paper reports high-fidelity RCS circuits that appear extremely challenging to simulate classically.The authors conclude H2’s RCS capability is strongly limited by qubit number rather than fidelity or clock speed.Source: §I · p. 2

II. THE H2 QUANTUM COMPUTER

H2 was upgraded from a 32-qubit to a 56-qubit configuration while retaining arbitrary connectivity. Primitive-operation fidelity did not degrade, although the larger system increased layer time and memory error.

II. THE H2 QUANTUM COMPUTER

  • The operational upgrade increased H2 from 32 to 56 qubits using reconfigured transport waveforms built from the same rearrangement primitives.The waveforms were developed to support a larger number of potential wells.Source: §II · p. 2
  • Transport 1Q randomized benchmarking with all 56 qubits captures combined memory and 1Q gate errors for the circuits studied.Operations use the bottom four of eight possible gate zones.Source: §II · p. 2
  • ϵ2Q improved from 1.83(5)×10−3 in Ref. to 1.57(5) × 10−3 in Table I.The passage attributes the improvement to technical changes including reduced laser phase noise.Source: §II · p. 2
  • At N = 56, a depth-1 layer took 80 ms and memory error was 4.0(4) × 10−4 per qubit per layer, versus 53 ms and 2.2(3) × 10−4 at N = 32.After Fig. 9 data collection, improved cooling reduced depth-1 time to around 70 ms and 2Q error to 1.28(8)×10−3.Source: §II · p. 2

III. THE DIFFICULTY OF RANDOM CIRCUIT SAMPLING

RCS tests sampling from quantum-circuit output distributions, but cross-entropy scores do not always establish faithful sampling and can be difficult to verify classically. The paper focuses on high-fidelity sampling from highly connected circuits and evaluates their classical simulation difficulty.

III. THE DIFFICULTY OF RANDOM CIRCUIT SAMPLING

  • H2 executes arbitrary-connectivity layers by relocating ions so each requested gate pair is co-located, then applying gates in batches across four active zones.For the illustrated layer, the two-qubit gates run in seven batches of four parallel gates.Source: Figure 1 · p. 3
  • Classical runtime for RCS depends on computer speed, qubit count, gate fidelity, and connectivity; prior demonstrations largely used 2D connectivity.The passage links limited connectivity beyond 2D to the technological difficulty of fixed-location platforms.Source: §III · p. 3
  • RCS compares classical and quantum computers by sampling bitstrings from a circuit’s output distribution, rather than reconstructing its quantum state.The task can be performed on either kind of computer.Source: §III · p. 3
  • The paper’s random-geometry circuits use perfect-entangler UZZ gates and Haar-random single-qubit gates, with N/2 two-qubit gates in each dense layer.The quantum computer is assumed to implement two-qubit gates with process infidelity ε in the preliminary model.Source: §III · pp. 3–4
  • A high cross-entropy score alone may not establish faithful sampling, because classical methods might attain such scores without reproducing the output distribution.Earlier RCS work generally assumed nonzero cross-entropy required a simulation of comparable fidelity.Source: §III · p. 4
  • Computing cross-entropy from quantum samples requires classical output-probability calculations, which are at least as hard as classical sampling in the target regime.The authors therefore report scores only for system sizes N ≤40 and also provide samples where those probabilities cannot be computed.Source: §III · p. 4
  • Linear cross-entropy benchmarking can track fidelity under weak noise, but in strong noise it can dramatically overestimate fidelity.For these random-geometry experiments, analysis and noisy simulations suggest the circuits are deeply in the weak-noise limit, subject to sufficient depth for Porter–Thomas probabilities.Source: §III · p. 4
  • Adversarial noise that removes gates can inflate cross-entropy relative to fidelity, but the authors argue H2’s high fidelity and connectivity make its circuits robust to such methods.Their circuit graphs have good edge expansion, so disconnecting equal-sized regions requires cutting an extensive number of gates, even at constant depth.Source: §III · p. 4

A. Circuit geometry and the cost of exact tensor-network contraction

Random-geometry circuits reach high tensor-network contraction complexity at lower depth than 2D circuits. The section defines complexity density and describes the circuit construction and contraction-cost estimates used for this comparison.

A. Circuit geometry and the cost of exact tensor-network contraction

  • The contraction cost is the minimum over tensor-contraction orders, estimated with cotengra heuristics targeting floating-point operations without memory constraints.Reported costs are upper bounds because the searches were not exhaustive; unconstrained memory may underestimate practical cost.Source: A. Circuit geometry and the cost of… · p. 5
  • For random geometries, exact-contraction cost continues growing rapidly with depth after complexity density saturates.Complexity density and full contraction cost therefore capture different parts of the cost behavior.Source: A. Circuit geometry and the cost of… · p. 5
  • Complexity density normalizes the effective qubit number, the approximate number of qubits whose worst-case simulation difficulty matches that of the circuit.Effective qubit number is related to the base-2 logarithm of the required floating-point operations per gate.Source: A. Circuit geometry and the cost of… · pp. 5–6
  • For spatially local circuits, fixed-depth complexity density vanishes asymptotically, so circuit depth must grow with system width to keep each qubit contributing to simulation complexity.The passage gives d ∼ N in 1D and d ∼ √N in 2D.Source: A. Circuit geometry and the cost of… · pp. 5–6
  • A random-geometry circuit samples a random d-regular graph, maps its vertices to qubits, and assigns edge colors to layers of perfect-entangler UZZ(π/2) gates.Haar-random single-qubit layers occur after initialization, before measurement, and between two-qubit layers.Source: Figure 2 · p. 5, A. Circuit geometry and the cost of… · p. 6
  • The 2D comparison uses square-lattice gate pairs and cyclically repeats four edge colors beyond depth four, matching circuits by total two-qubit gate count.The 2D circuits are slightly less densely gated; the reported cost estimates use UZZ(π/2), not iSWAP-like gates.Source: A. Circuit geometry and the cost of… · pp. 6–7
  • At N = 56, complexity density saturates near unity at roughly d ≈12 for random geometries versus d ≈22 for 2D; the figure compares both geometries across N and depth.Saturation corresponds to contraction cost approaching brute-force 56-qubit state-vector simulation cost.Source: Figure 3 · p. 6, A. Circuit geometry and the cost of… · p. 7
  • Random-geometry circuit complexity density remains constant at fixed depth as N grows, whereas 2D circuits require d ∼ N to maintain it.The paper attributes the random-geometry behavior to gate pairs drawn from graphs with good asymptotic expansion properties.Source: Figure 3 · p. 6, A. Circuit geometry and the cost of… · p. 7

B. Exact contraction cost with memory constraints

The section compares optimized exact tensor-network contraction paths with and without memory constraints. Slicing limits contraction width but becomes exponentially costly at greater depths, reaching a median of ~10^30 FLOPs at d=20.

C. The cost of approximate tensor network simulations

The section evaluates DMRG-based MPS simulation of 2D and random-geometry circuits. At depth 20, matching H2's gate error for random geometries appears to require an essentially exact MPS, while random geometry also raises the cost of reaching a fixed error.

D. Time requirements for verifiable RCS

Verifiable RCS requires enough samples to distinguish its output from randomness, so the section relates total run time to per-shot time and fidelity. It compares the effective qubit numbers achievable in 2D and random geometries under fixed run-time and gate-error assumptions.

D. Time requirements for verifiable RCS

  • The uncertainty in FXEB decreases as the square root of the sample count increases, setting the data requirement for verifiable RCS.For a fixed total run time, the required number of samples also depends on the time for one circuit shot.Source: D. Time requirements for verifiable RCS · p. 10
  • Nmax is evaluated across assumed 2Q gate-error rates and circuit times, with progressively faster circuit-time assumptions shown above the 1-second curves.The figure compares 2D and random geometries over a fixed 100 seconds of data.Source: Figure 7 · p. 10, D. Time requirements for verifiable RCS · p. 11
  • Random geometries yield larger effective qubit numbers at fixed gate error and circuit time because their classical simulation cost saturates quickly with depth.The comparison maximizes effective qubit number over circuit depths within a fixed 100-second total run time.Source: Figure 7 · p. 10, D. Time requirements for verifiable RCS · p. 11

IV. THE FIDELITY OF RANDOM-GEOMETRY CIRCUITS ON H2

The section estimates random-geometry circuit fidelities with mirror benchmarking and, where feasible, linear cross-entropy benchmarking, alongside a gate-counting model. The measurements show high fidelities and agreement between estimates in the classically verifiable range.

IV. THE FIDELITY OF RANDOM-GEOMETRY CIRCUITS ON H2

  • Mirror benchmarking estimates depth-d fidelity from the all-qubits return probability of a mirrored depth-d circuit, while FXEB is computed directly for circuits up to 40 qubits.Randomized compiling on the mirrored half prevents potential cancellation of coherent errors; FXEB is not computed for the 56-qubit data.Source: §IV · p. 11
  • The gate-counting model combines 2Q-gate and memory errors, but its fidelity estimates are systematically low and its effective depth is shifted by about 1.12 for comparison.Memory error is measured separately because it varies with circuit geometry and qubit number.Source: §IV · pp. 11–12
  • RCS fidelities are roughly two orders of magnitude above prior demonstrations on classically difficult circuits, with H2 mirror-benchmarked at 56 qubits through depth 20.Depth-24 mirror circuits also provide an estimate for the depth-12, 56-qubit circuits.Source: §IV · pp. 12–13, Figure 9 · p. 13
  • At N=40 and depth 12, FXEB agrees well with FMB and the gate-counting model across qubit numbers; a depth sweep also compares FXEB and FMB.The experiments used 50 circuit randomizations and 20 shots per randomization, except the depth-24 experiment, which used 100 shots per randomization.Source: Figure 9 · p. 13, §IV · p. 13

V. OUTLOOK

The outlook notes that the empirical difficulty of noisy RCS remains uncertain across experimentally accessible sizes and error rates. It argues that continued scaling while reducing errors could extend non-error-corrected demonstrations into more classically challenging regimes.

V. OUTLOOK

  • Noisy-RCS asymptotics remain comparatively poorly understood, and existing results do not determine difficulty across experimental system sizes and error rates.The cited easiness result assumes a particular noise structure and circuit depth.Source: §V · p. 13
  • H2’s high gate fidelities and arbitrary connectivity support RCS in a computationally challenging regime at unprecedented fidelities.The authors see room to scale these demonstrations without further reducing gate error rates.Source: §V · p. 14
  • Because random-geometry circuits need low depths to become hard to simulate, near-term QCCD scaling could enable faithful circuits beyond conceivable classical calculations.The outlook identifies larger system sizes, lower memory errors, shorter circuit times, and lower 2Q error rates as achievable directions.Source: §V · p. 14

A1. PRIMITIVE OPERATION BENCHMARKS AT N = 56

The appendix reports SPAM and single-qubit randomized-benchmarking measurements on the H2 gate-zone qubits.

A. State-preparation and measurement (SPAM) experiment

  • The average SPAM error is 1.47(9)×10^-3, calculated from the unequal p(1|0) and p(0|1) error probabilities.The reported probabilities are p(1|0)=3.8(7)×10^-4 and p(0|1)=26(2)×10^-4.Source: A. State-preparation and measurement… · p. 15

B. 1Q randomized benchmarking (1Q RB)

  • The average 1Q Clifford infidelity is 2.9(4)×10^-5 from randomized benchmarking across the eight gate-zone qubits.The RB sequences used lengths 2, 512, and 2048, with four randomizations and 100 shots per randomization.Source: B. 1Q randomized benchmarking (1Q RB) · p. 15

C. 2Q randomized benchmarking (2Q RB)

Two-qubit randomized benchmarking across H2’s four gate zones estimates an average two-qubit gate infidelity of 1.57(5) × 10^-3. The benchmark combines measurements from all four gate-zone qubit pairs.

C. 2Q randomized benchmarking (2Q RB)

D. Transport one-qubit randomized benchmarking (Transport 1Q RB)

Transport 1Q RB measures errors across up to 56 qubits while accounting for memory errors during transport. The section also models how memory error varies with qubit number and examines circuit timing, fidelity estimators, and convergence to Porter-Thomas output probabilities.

D. Transport one-qubit randomized benchmarking (Transport 1Q RB)

  • Transport 1Q RB across N = 16, 24, 32, 40, 48, and 56 estimates per-qubit error from memory errors during random transport and 1Q gates.The experiment used random SU(2) unitaries interleaved with transport operations designed to mimic the main-text circuits.Source: D. Transport one-qubit randomized… · p. 16, Figure A4 · p. 17
  • The memory-error model is logistic, interpolating between zero error and a maximum as qubit number rises; its fitted parameters are A = 4.1(2) × 10^-4, N0 = 20(2), and k = 0.18(5).The interpolated memory-error estimates combine with other component benchmarks to produce gate-counting fidelity estimates as a function of N.Source: D. Transport one-qubit randomized… · pp. 16–17, Table A1 · p. 16
  • RCS and transport 1Q RB have essentially identical shot times, while mirror-benchmarking circuits show roughly (d − 2)/(d − 1) scaling plus about 2.2% overhead.For transport 1Q RB at N = 56, time per shot is essentially linear with depth; the fitted model is time = (0.080d + 0.019) s.Source: D. Transport one-qubit randomized… · p. 17, Figure A5 · p. 17, Figure A6 · p. 17

A2. COMPARISON OF VARIOUS FIDELITY ESTIMATORS

  • For sufficiently deep circuits, the authors expect linear XEB fidelity to agree with gate-counting and mirror-benchmarking estimates, and test this using simulations that hold the 2Q-gate failure probability per layer fixed across system sizes.Mirror benchmarking uses the return probability after a depth-d/2 circuit followed by its inverted reverse; the resulting estimate is denoted FMB.Source: A2. COMPARISON OF VARIOUS FIDELITY… · p. 18

A. Convergence of RG circuits to Porter-Thomas without noise

  • By depth 10, output probabilities averaged over 100 circuits converge well to Porter-Thomas for N = 16; at N = 28, a single circuit also converges well.At depths d ≥ 10, most of the 100-circuit distribution has |FXEB − 1| fluctuations well below 1%.Source: A. Convergence of RG circuits to… · p. 18, Figure A7 · p. 18
  • At insufficient circuit depths, output probabilities are not Porter-Thomas distributed even without noise, so linear XEB is not a good fidelity estimator.The convergence assessment covers both second-moment deviations and the full distribution of binned output probabilities.Source: A. Convergence of RG circuits to… · p. 18

B. Numerical studies with a detailed noise model

Numerical studies compare fidelity estimators under a detailed noise model and find that their estimates track computed circuit fidelity reasonably well. The simulations also expose systematic boundary-related differences and support the expected similarity of estimates across simulated system sizes.

B. Numerical studies with a detailed noise model

  • The simulation model combines stochastic Pauli channels on 2Q gates with memory errors during idling, assuming memory error is entirely coherent dephasing.Rates are rescaled by 56/N so the error rate per 2Q-gate layer remains fixed as N changes.Source: B. Numerical studies with a detailed… · pp. 18–19
  • By depth 10, the output-probability distribution is well-converged to Porter-Thomas for N = 16 averaged over 100 circuits and for a single circuit at N = 28.The plotted probabilities are rescaled by 2^N.Source: Figure A8 · p. 19
  • Fidelity-estimator data collapse across simulated system sizes, suggesting that their similarity should also hold at N = 56.At N = 24, the largest simulated size, all three estimates agree reasonably well with computed fidelity, with small systematic differences.Source: B. Numerical studies with a detailed… · p. 19, Figure A9 · p. 20, Figure A10 · p. 20
  • Boundary errors cause gate counting to underestimate true fidelity, while mirror benchmarking and XEB overestimate it through imperfect state orthogonalization.The authors fit gate-counting fidelity at shifted depth d → d − δ to the N = 24 mirror-benchmarking numerics, obtaining δ ≈ 1.12.Source: B. Numerical studies with a detailed… · pp. 19–20

A3. QUANTIFYING UNCERTAINTIES IN THE EXPERIMENTAL DATA

The appendix explains how resampling choices affect uncertainty estimates, then describes the construction of random-geometry circuits and their tensor-network representation.

A3. QUANTIFYING UNCERTAINTIES IN THE EXPERIMENTAL DATA

  • Aggregate resampling pools observed shots, whereas double resampling resamples circuits and then their shots; the paper uses aggregate resampling for all reported 1σ error bars.Double resampling generally produces wider confidence intervals.Source: A3. QUANTIFYING UNCERTAINTIES IN THE… · pp. 21–22
  • For expected fluctuations µ <∼0.2, aggregate resampling gives adequate FXEB coverage, while FMB requires µ <∼0.1; double resampling generally over-covers.At µ = 1, aggregate resampling under-covers both measures, while double resampling remains adequate.Source: A3. QUANTIFYING UNCERTAINTIES IN THE… · p. 21, Figure A12 · p. 22

A4. CONSTRUCTION OF CIRCUITS WITH RANDOM GEOMETRIES

  • QCCD can execute arbitrary qubit pairs in a two-qubit layer by physically rearranging ions, without logical SWAP gates.Source: A4. CONSTRUCTION OF CIRCUITS WITH… · p. 23
  • Random-geometry circuits are built from random d-regular graphs whose properly colored edges define d disjoint two-qubit gate layers.The construction assumes a proper d-coloring can be found; graphs for which one is not found are rejected and resampled.Source: A4. CONSTRUCTION OF CIRCUITS WITH… · p. 23

A5. BOUNDS ON CONTRACTION DIFFICULTY OF RANDOM-GEOMETRY CIRCUITS

A. Lower bound

This section characterizes lower bounds and costs for tensor-network and approximate MPS simulation of random-geometry circuits, including how depth, gate rank, optimizer choice, and blocking strategies affect contraction or simulation costs. It also examines the bond dimensions needed to match experimental fidelities and the sampling overhead of post-processing to boost XEB, alongside Clifford-based bounds.

A. Lower bound

  • For d ≥ 3, with probability approaching 1 as N grows, no contraction ordering avoids an intermediate tensor with rank linear in N.The proof uses expansion of random regular graphs: extensive boundaries force intermediate tensors with many legs. It bounds the decomposed network TC, whose contraction cost lower-bounds that of the original circuit network because its gates can be regrouped.Source: A. Lower bound · pp. 24–25

B. Upper bound

  • The authors prove that 1 − Cd cannot vanish abruptly as depth increases, while exponential decay remains possible but unproven.The bound comes from an explicit contraction ordering based on output-pair causal cones and qubit reuse.Source: B. Upper bound · p. 27

A6. SIMPLE MODELS OF EXACT CONTRACTION COST

  • For 2D circuits, the fitted envelope f(˜d) = 1.1 min(1.0, 0.35 ˜d) qualitatively matches cotengra costs; for random-geometry circuits, Cd,N ≈ a × min(1, c(d − 2)) with (a, c) = (1, 0.125) also agrees well.The 2D model collapses complexity density onto a universal curve at large N; the random-geometry density rises roughly linearly until d ≈ 10, then quickly saturates to 1.Source: A6. SIMPLE MODELS OF EXACT CONTRACTION… · p. 28

A7. IMPACT OF GATE CHOICE ON SIMULATION HARDNESS

  • On a 1D circuit, rank-4 gates give sideways contraction cost 2^d, while rank-2 gates give 2^(d/2), so rank-4 gates effectively double depth at equal cost.This example illustrates why lower-rank gate decompositions can help at low depths, when optimal contraction need not proceed in a time-like direction.Source: A7. IMPACT OF GATE CHOICE ON… · p. 29
  • At fixed circuit depth, rank-4 gates should be preferred because they lead to higher contraction costs; Quantinuum’s native UZZ(θ) gate is not rank 4, and building a rank-4 gate requires at least two native gates.Source: A7. IMPACT OF GATE CHOICE ON… · p. 29

A8. COMPARISON OF VARIOUS CONTRACTION-ORDER OPTIMIZERS

  • For fixed optimization time, cuQuantum and cotengra generally outperform tamaki and multilevel; cotengra is generally competitive with cuQuantum when both optimize for more than about 100 seconds.The cost comparison used 10 random 56-qubit, depth-10 circuit instances and compared networks without simplification.Source: A8. COMPARISON OF VARIOUS… · pp. 29–30

A9. ACHIEVABLE FIDELITIES WITH DMRG

  • Blocking groups qubits into MPS tensors and uses hypergraph partitioning to minimize inter-block gates, which can improve fidelity at fixed resources when aligned with inhomogeneous entanglement.The authors found that adding even one gate layer to the middle circuit segment dramatically increased contraction cost for random geometries.Source: A9. ACHIEVABLE FIDELITIES WITH DMRG · p. 31, Figure A23 · p. 32
  • Extrapolated DMRG requires bond dimensions in excess of 10^5 at depth 10 and 10^7 at depth 20 to match experimental error rates.The estimates depend only weakly on blocking, and extrapolating from small bond dimensions may bias the required dimensions downward.Source: A9. ACHIEVABLE FIDELITIES WITH DMRG · pp. 31–32, Figure A24 · p. 32
  • For 24-qubit, depth-10 circuits, MPS fidelity F closely tracks achievable FXEB when amplitudes are estimated by evolving input and sampled bitstrings to the circuit midpoint.The analysis uses direct SVD approximations and 1,000 bitstrings for each of 100 random circuits.Source: A9. ACHIEVABLE FIDELITIES WITH DMRG · pp. 32–33
  • Post-processing estimated amplitudes can raise FXEB above simulation fidelity F, but achieving a multiplicative enhancement requires the retained bitstring fraction to shrink exponentially.The authors therefore see no obvious better strategy, on average across circuits, than choosing a bond dimension that makes F comparable to FXEB.Source: A9. ACHIEVABLE FIDELITIES WITH DMRG · p. 33, Figure A25 · p. 33
  • Clifford purity simulations provide lower bounds on MPS bond dimension, but practical DMRG does not saturate them and requires dimensions near 10^5 even at depth 10.The bound is tight only for flat Schmidt spectra, while greedy DMRG may miss the globally optimal MPS.Source: A9. ACHIEVABLE FIDELITIES WITH DMRG · p. 33, Figure A26 · p. 33

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