Divide-and-conquer verification method for noisy intermediate-scale quantum computation
At a Glance
Section titled āAt a Glanceā| Metadata | Details |
|---|---|
| Publication Date | 2022-07-07 |
| Journal | Quantum |
| Authors | Yuki Takeuchi, Yasuhiro Takahashi, Tomoyuki Morimae, Seiichiro Tani |
| Institutions | Gunma University, Kyoto University |
| Citations | 8 |
| Analysis | Full AI Review Included |
Technical Documentation: Efficient Verification for NISQ Computation using Divide-and-Conquer Methods
Section titled āTechnical Documentation: Efficient Verification for NISQ Computation using Divide-and-Conquer MethodsāReference Paper: Divide-and-conquer verification method for noisy intermediate-scale quantum computation (Takeuchi et al., arXiv:2109.14928v3)
Executive Summary
Section titled āExecutive SummaryāThis research introduces a highly efficient method for verifying the output fidelity of Noisy Intermediate-Scale Quantum (NISQ) computations, a critical challenge for current quantum hardware development.
- Core Achievement: Proposes a divide-and-conquer verification protocol that drastically reduces the sample complexity required to estimate the fidelity ($\langle\psi_t|\rho_{out}|\psi_t\rangle$).
- Complexity Reduction: The method achieves polynomial sample complexity, $O(2^{12D} / \epsilon^6)$, where $D$ is the circuit denseness ($D = O(\log n)$ for sparse chips). This overcomes the exponential $O(2^n)$ requirement of direct fidelity estimation.
- Target Application: Specifically designed for shallow (logarithmic-depth) quantum circuits implemented on sparse quantum computing chips (e.g., planar graph architectures).
- Methodology: Utilizes generalized stabilizer operators and decomposes the verification circuit into smaller, manageable (m+1)-qubit and (n-m+1)-qubit measurement circuits.
- Hardware Requirement: The protocol demands extremely high-fidelity quantum gates, requiring the diamond norm distance between ideal and actual gates to be bounded by $\epsilon/4^{D+2}$.
- Validation: Proof-of-principle experiment successfully performed on the IBM Manila 5-qubit superconducting chip, demonstrating practical performance.
Technical Specifications
Section titled āTechnical SpecificationsāThe following hard data points summarize the performance and constraints derived from the verification protocol:
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Target Fidelity Accuracy (ε) | 0.1 | Dimensionless | Desired verification accuracy. |
| Required Gate Precision (Diamond Norm) | < ε / 4D+2 | Dimensionless | Upper bound on error for (m+1)-qubit gates, critical for protocol success. |
| Worst-Case Sample Complexity (Copies of ρout) | O(212D / ε6) | Copies | Required number of output state copies for verification. D = O(log n). |
| Experimental Qubit Count (n) | 4 | Qubits | Used for proof-of-principle experiment on IBM Manila. |
| Experimental Measurement Repetitions (T) | 1024 | Times | Repetitions for estimating each term in the fidelity calculation. |
| Experimental Fidelity Estimate (Fest) | ~ 0.789 | Dimensionless | Fidelity obtained using the divide-and-conquer method. |
| Direct Fidelity Estimate (Fāest) | ~ 0.865 | Dimensionless | Fidelity obtained using direct Pauli measurement comparison. |
| Fidelity Difference ( | Fest - Fāest | ) | < 0.076 |
Key Methodologies
Section titled āKey MethodologiesāThe verification method, summarized in Algorithm 1, relies on spatial and temporal decomposition of the quantum circuit:
- Qubit Partitioning (Spatial Division): The n-qubit system is divided into two sets, m and (n - m) qubits, minimizing the number of Controlled-Z (CZ) gates crossing the boundary (Denseness D).
- Unitary Decomposition: The target quantum circuit U is decomposed into a linear combination of tensor products of smaller m-qubit and (n - m)-qubit operators (U = Σ Qij).
- Generalized Stabilizer Operators: The fidelity $\langle\psi_t|\rho_{out}|\psi_t\rangle$ is estimated by measuring the expectation value of generalized stabilizer operators ŝk, which are products of gi = UZiU† operators.
- Time Division Technique: The complex (n+1)-qubit measurement circuit required for estimating Tr[ρoutŝk] is replaced by two smaller, independent (m+1)-qubit and (n-m+1)-qubit circuits.
- Identity Gate Replacement: This division is achieved by replacing the identity gate connecting the two sub-circuits with a combination of Pauli-basis measurements and preparations (Clifford operations Cl).
- Classical Post-Processing: The measurement outcomes from the smaller circuits are classically post-processed (averaged T1, T2, T3 times) to yield the final estimated fidelity Fest.
6CCVD Solutions & Capabilities
Section titled ā6CCVD Solutions & CapabilitiesāThe efficient verification of NISQ devices, as demonstrated in this paper, places extreme demands on the underlying quantum hardwareāspecifically requiring ultra-low noise and high-fidelity gates. While the experiment used superconducting qubits, 6CCVDās MPCVD diamond materials are essential for advancing solid-state quantum platforms (such as NV centers in diamond or SiC defects) that compete in the NISQ era.
Achieving the required gate precision (diamond norm < ε/4D+2) necessitates materials with unparalleled purity and surface quality, which 6CCVD delivers.
| Research Requirement | 6CCVD Material Solution | Technical Specification & Value Proposition |
|---|---|---|
| Ultra-High Coherence & Low Noise | Optical Grade Single Crystal Diamond (SCD) | Essential for solid-state qubits (NV, SiC). 6CCVD provides SCD with ultra-low nitrogen concentration (< 1 ppb) and minimal strain, maximizing T2 coherence times necessary for high-fidelity, logarithmic-depth circuits. |
| Precision Substrate Dimensions | Custom SCD/PCD Plates | We offer custom dimensions for plates and wafers up to 125mm (PCD). SCD thicknesses range from 0.1Āµm to 500Āµm, allowing integration into complex quantum chip architectures. |
| Minimizing Surface Defects | Ultra-Smooth Polishing | Verification protocols are highly sensitive to gate errors. 6CCVD guarantees surface roughness (Ra) < 1nm for SCD and < 5nm for inch-size PCD, critical for minimizing surface noise and enabling high-resolution lithography. |
| Integrated Control & Measurement | In-House Metalization Services | The verification method requires precise control and measurement circuitry. We offer custom deposition of Au, Pt, Pd, Ti, W, and Cu, supporting the integration of microwave lines or superconducting contacts directly onto the diamond substrate. |
| Conductive Diamond Applications | Boron-Doped Diamond (BDD) | For future extensions of verification methods into quantum sensing or electrochemical applications, 6CCVD supplies BDD with tailored doping levels and thicknesses (0.1Āµm - 500Āµm). |
| Complex Engineering Support | Expert Consultation & Global Logistics | Our in-house PhD team provides authoritative engineering support for material selection, orientation, and surface preparation to meet the stringent requirements of NISQ verification projects. Global shipping (DDU default, DDP available) ensures timely delivery worldwide. |
For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.
View Original Abstract
Several noisy intermediate-scale quantum computations can be regarded as logarithmic-depth quantum circuits on a sparse quantum computing chip, where two-qubit gates can be directly applied on only some pairs of qubits. In this paper, we propose a method to efficiently verify such noisy intermediate-scale quantum computation. To this end, we first characterize small-scale quantum operations with respect to the diamond norm. Then by using these characterized quantum operations, we estimate the fidelity <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mo fence=āfalseā stretchy=āfalseā>&#x27E8;</mml:mo><mml:msub><mml:mi>&#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mo stretchy=āfalseā>|</mml:mo></mml:mrow><mml:msub><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mover><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy=āfalseā>&#x005E;</mml:mo></mml:mover></mml:mrow><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mi mathvariant=ānormalā>o</mml:mi><mml:mi mathvariant=ānormalā>u</mml:mi><mml:mi mathvariant=ānormalā>t</mml:mi></mml:mrow></mml:msub><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mo stretchy=āfalseā>|</mml:mo></mml:mrow><mml:msub><mml:mi>&#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo fence=āfalseā stretchy=āfalseā>&#x27E9;</mml:mo></mml:math> between an actual <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>n</mml:mi></mml:math>-qubit output state <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:msub><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mover><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy=āfalseā>&#x005E;</mml:mo></mml:mover></mml:mrow><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mi mathvariant=ānormalā>o</mml:mi><mml:mi mathvariant=ānormalā>u</mml:mi><mml:mi mathvariant=ānormalā>t</mml:mi></mml:mrow></mml:msub></mml:math> obtained from the noisy intermediate-scale quantum computation and the ideal output state (i.e., the target state) <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mo stretchy=āfalseā>|</mml:mo></mml:mrow><mml:msub><mml:mi>&#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo fence=āfalseā stretchy=āfalseā>&#x27E9;</mml:mo></mml:math>. Although the direct fidelity estimation method requires <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>O</mml:mi><mml:mo stretchy=āfalseā>(</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup><mml:mo stretchy=āfalseā>)</mml:mo></mml:math> copies of <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:msub><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mover><mml:mi>&#x03C1;</mml:mi><mml:mo stretchy=āfalseā>&#x005E;</mml:mo></mml:mover></mml:mrow><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mi mathvariant=ānormalā>o</mml:mi><mml:mi mathvariant=ānormalā>u</mml:mi><mml:mi mathvariant=ānormalā>t</mml:mi></mml:mrow></mml:msub></mml:math> on average, our method requires only <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>O</mml:mi><mml:mo stretchy=āfalseā>(</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mn>2</mml:mn><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mn>12</mml:mn><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=āfalseā>)</mml:mo></mml:math> copies even in the worst case, where <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>D</mml:mi></mml:math> is the denseness of <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mo stretchy=āfalseā>|</mml:mo></mml:mrow><mml:msub><mml:mi>&#x03C8;</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo fence=āfalseā stretchy=āfalseā>&#x27E9;</mml:mo></mml:math>. For logarithmic-depth quantum circuits on a sparse chip, <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>D</mml:mi></mml:math> is at most <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>O</mml:mi><mml:mo stretchy=āfalseā>(</mml:mo><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mi>n</mml:mi></mml:mrow><mml:mo stretchy=āfalseā>)</mml:mo></mml:math>, and thus <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>O</mml:mi><mml:mo stretchy=āfalseā>(</mml:mo><mml:msup><mml:mi>D</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mn>2</mml:mn><mml:mrow class=āMJX-TeXAtom-ORDā><mml:mn>12</mml:mn><mml:mi>D</mml:mi></mml:mrow></mml:msup><mml:mo stretchy=āfalseā>)</mml:mo></mml:math> is a polynomial in <mml:math xmlns:mml=āhttp://www.w3.org/1998/Math/MathMLā><mml:mi>n</mml:mi></mml:math>. By using the IBM Manila 5-qubit chip, we also perform a proof-of-principle experiment to observe the practical performance of our method.