Gate-error analysis in simulations of quantum computers with transmon qubits

At a Glance

Metadata	Details
Publication Date	2017-12-01
Journal	Physical review. A/Physical review, A
Authors	Dennis Willsch, Madita Nocon, Fengping Jin, Hans De Raedt, Kristel Michielsen
Institutions	RWTH Aachen University, University of Groningen
Citations	52
Analysis	Full AI Review Included

6CCVD Technical Analysis & Documentation

Executive Summary: High-Coherence Diamond for Superconducting Qubit Architectures

This analysis of the superconducting transmon qubit simulations highlights the critical need for ultra-high-purity materials and precision engineering to overcome fundamental system leakage errors, particularly as observed in echoed cross-resonance (CR) gates.

Core Challenge: Even in idealized simulations using the time-dependent Schrödinger equation (TDSE), two-qubit CNOT gates exhibit high Diamond Error Rates ($\eta_{\diamond} > 2%$). This systemic error stems from leakage into non-computational states (higher transmon levels and resonator photons).
Material Implication: Leakage and decoherence are inherently linked to the substrate quality, particularly minimizing two-level systems (TLS) at the material-superconductor interface. Ultra-high-purity, defect-free substrates are essential for achieving the required fault-tolerance thresholds.
Performance Metrics: Optimized gates achieved high Average Gate Fidelities ($F_{avg} \approx 0.995$), but the high Diamond Distance ($\eta_{\diamond}$) indicates non-Pauli errors, which are more detrimental to fault-tolerant computation.
Gate Pulse Requirements: Implementation requires high-precision microwave Gaussian and flat-topped Gaussian pulses (duration up to 653 ns) and advanced error mitigation techniques (DRAG, VZ phase correction).
Simulation vs. Application: The study confirms that gate metrics alone cannot reliably predict performance in complex quantum circuits (e.g., Quantum Fourier Transform, QFT), underscoring the necessity of real-world testing on high-quality substrates.
6CCVD Value Proposition: 6CCVD’s specialized Single Crystal Diamond (SCD) and Polycrystalline Diamond (PCD) substrates, combined with precision polishing (Ra < 1 nm) and custom thin-film metalization, are the ideal foundation for building high-coherence superconducting quantum processors capable of meeting stringent fault-tolerance requirements.

Technical Specifications

The following parameters summarize the key findings regarding the simulated transmon system architecture and the optimized two-qubit CNOT gate performance metrics.

Parameter	Value	Unit	Context
Resonator Frequency	7.0	GHz	Reference frequency ($\omega_r/2\pi$)
Qubit 1 Frequency (Shifted)	5.346	GHz	Measured locally rotating frame ($\tilde{\omega}_1/2\pi$)
Qubit 2 Frequency (Shifted)	5.118	GHz	Measured locally rotating frame ($\tilde{\omega}_2/2\pi$)
Single-Qubit Gate Duration	83	ns	Standard Gaussian envelope (X_$\pi/2$, X_$\pi$)
Longest CNOT Pulse Duration ($T_{CR}$)	652.954	ns	CR4 CNOT₁₂ (includes 30 ns rise/fall time)
Best Average Gate Fidelity ($F_{avg}$)	0.9951	-	Achieved by CR1 CNOT₂₁
Worst Average Gate Fidelity ($F_{avg}$)	0.9842	-	Observed for CR1 CNOT₁₂
Diamond Error Rate ($\eta_{\diamond}$) Range	0.020 - 0.049	-	All gates showed $\eta_{\diamond} > 2%$, indicating persistent non-Pauli errors (leakage).
Best Unitarity ($u$)	0.992	-	Achieved by CR2 CNOT₂₁
Worst Unitarity ($u$)	0.969	-	Observed for CR1 CNOT₁₂
Goal Fidelity for Fault-Tolerance	0.999995	-	Required to qualify with known quantum error-correcting codes.

Key Methodologies

The study relies on detailed, non-perturbative simulation of the quantum system dynamics, focusing on generating control pulses that mimic experimental protocols (IBM Quantum Experience architecture).

Simulation Engine: Solving the time-dependent Schrödinger equation (TDSE) for a generic model Hamiltonian of two transmons coupled by a resonator.
Algorithm Implementation: Utilizing a robust Suzuki-Trotter product-formula algorithm implemented in C++ to achieve unconditional stability and inherent unitarity in the total Hilbert space simulation.
Modeling Physical Gates: Quantum gates are implemented via sequences of microwave voltage pulses, mathematically modeled as generic sums of shaped envelopes:
- Single-Qubit Pulses: Gaussian envelopes with DRAG (Derivative Removal by Adiabatic Gate) correction applied to mitigate leakage to higher transmon levels (e.g., $m_i=2$).
- Two-Qubit Pulses (CNOT): Implemented using the cross-resonance (CR) effect, utilizing flat-topped Gaussian pulses (30 ns rise time) oscillating at the target qubit frequency.
- Echo Schemes: Comparison of three CNOT schemes: CR1 (one-pulse), CR2 (two-pulse echoed), and CR4 (four-pulse echoed).
Pulse Optimization: Pulse parameters (amplitude $\Omega_0$, DRAG coefficient $\beta$, phase $\gamma$, pulse time $T$) were optimized using a Nelder and Mead multidimensional scheme to minimize the discrepancy $\Delta(M, U)$ between the simulated transformation matrix $M$ and the ideal unitary $U$.
Error Correction Modeling: Incorporation of Virtual Z (VZ) gates—implemented by shifting the phase of subsequent pulses—to compensate for phase errors resulting from off-resonant driving.

6CCVD Solutions & Capabilities

The core finding of this research—that systemic errors persist even in idealized simulations due to non-computational leakage—underscores the need for materials that fundamentally minimize decoherence and parasitic coupling mechanisms. 6CCVD provides the specialized diamond substrates and precision processing required to push superconducting qubits past the fault-tolerance threshold ($\eta_{\diamond} \ll 1%$).

Applicable Materials: The Foundation for High Coherence

The pursuit of ultra-low leakage and high fidelity in transmon systems requires substrates optimized for minimal surface defects (reducing Two-Level Systems, TLS) and high thermal management.

Optical Grade SCD (Single Crystal Diamond):
- Requirement: Ultra-low surface roughness and high purity (low nitrogen/defects) are critical to prevent interface loss mechanisms—the physical manifestation of the modeled computational leakage.
- 6CCVD Solution: Our SCD wafers provide the highest purity and thermal conductivity, making them ideal for high-coherence research environments, offering a path to reduce the systemic errors observed in the TDSE simulations.
High-Purity PCD (Polycrystalline Diamond) Substrates:
- Requirement: Large-area wafers for complex, scaled-up quantum architectures (which typically involve dozens or hundreds of qubits and resonators).
- 6CCVD Solution: We offer PCD plates/wafers up to 125 mm in diameter, supporting large-scale processor development required for testing complex circuits like the QFT.

Advanced Customization Capabilities

The precise implementation of transmon gates relies on micro-scale fabrication of Josephson junctions and control wiring. 6CCVD provides end-to-end processing support for research needs:

Capability	Technical Specification	Relevance to Qubit Architecture
Custom Dimensions	Plates/wafers up to 125 mm (PCD). Substrates up to 10 mm thick.	Accommodates large-scale circuits and high-power applications (e.g., resonator coupling design).
Material Thickness Control	SCD/PCD from 0.1 µm up to 500 µm (wafers), Substrates up to 10 mm.	Essential for optimizing material stack integration and achieving precise resonant frequencies.
Precision Polishing	Ra < 1 nm (SCD), Ra < 5 nm (Inch-size PCD).	Directly addresses the leakage/decoherence problem by minimizing surface roughness and reducing surface TLS density.
Custom Metalization	Deposition of Au, Pt, Pd, Ti, W, Cu films.	Supports the fabrication of complex superconducting circuits (e.g., Al/Nb films) and control electrodes required for applying the Gaussian microwave pulses modeled in the paper.
Laser Cutting Services	Precision dicing and custom shape generation.	Enables the production of unique chip geometries and integration features necessary for coupling qubits to external control lines and cryogenic packaging.

Engineering Support

The observed discrepancy between high gate fidelity ($F_{avg}$) and poor circuit performance (seen in the repeated gate applications section) confirms that material choice and engineering design are critical. 6CCVD’s in-house PhD team can assist with material selection and specification for projects involving superconducting transmon qubit architectures or similar high-coherence solid-state quantum systems.

For researchers focused on minimizing the non-Pauli leakage errors identified in this simulation, 6CCVD provides materials engineered for the highest possible purity and surface quality.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

In the model of gate-based quantum computation, the qubits are controlled by a sequence of quantum gates. In superconducting qubit systems, these gates can be implemented by voltage pulses. The success of implementing a particular gate can be expressed by various metrics such as the average gate fidelity, the diamond distance, and the unitarity. We analyze these metrics of gate pulses for a system of two superconducting transmon qubits coupled by a resonator, a system inspired by the architecture of the IBM Quantum Experience. The metrics are obtained by numerical solution of the time-dependent Schrodinger equation of the transmon system. We find that the metrics reflect systematic errors that are most pronounced for echoed cross-resonance gates, but that none of the studied metrics can reliably predict the performance of a gate when used repeatedly in a quantum algorithm.

Tech Support

Original Source

DOI: https://doi.org/10.1103/physreva.96.062302