Accuracy-Cost Trade-offs for Reference VQE Calculations of H$_2$ on IBM Quantum Hardware

This paper presents a hardware-validated reference dataset and analysis of H$_2$ ground-state energy calculations on 2026 IBM Quantum processors, demonstrating that circuit simplification via tapered mappings offers the most consistent accuracy gains while resilience level 1 incurs substantial costs and session-based execution provides no systematic accuracy advantage despite higher billed time.

The Gist

Imagine you are trying to bake the perfect chocolate cake (finding the exact energy of a hydrogen molecule) using a very new, slightly glitchy oven (a quantum computer). You want the cake to taste perfect, but you also don't want to spend a fortune on electricity or wait all day for it to bake.

This paper is essentially a consumer report for people trying to bake cakes on IBM's new quantum ovens in 2026. The authors tested every possible setting—how long to bake, which oven to use, how to mix the ingredients, and whether to keep the oven door closed or open between batches—to see what gives the best cake for the lowest price.

Here is the breakdown of their findings using simple analogies:

1. The Setup: The "Glitchy" Oven

The researchers used the Hydrogen molecule (H2) as their test cake. It's the simplest possible cake (just two ingredients), so they know exactly what the perfect result should be. This allowed them to see exactly how much the "glitchy" oven messed up the recipe.

They tested this on several different IBM quantum computers (the "ovens"), ranging from older models to the newest, fastest ones.

2. The Big Findings (The "Secrets" to a Good Cake)

A. The Recipe Matters More Than the Oven (Circuit Mapping)

• The Analogy: Imagine you have a recipe that says "mix 100 bowls of flour" versus one that says "mix 1 cup of flour." If your kitchen is messy (noisy), mixing 100 bowls gives you a 99% chance of spilling flour everywhere. Mixing 1 cup gives you a much better chance of a clean cake.
• The Result: The most important thing they found was simplifying the recipe. By using a specific mathematical trick to shrink the problem down (called "tapering"), they reduced the number of "steps" the computer had to take.
  • Outcome: Smaller, simpler circuits (recipes) consistently produced better results and were cheaper to run, even on the same computer. It didn't matter if you used the fancy new oven or the old one; a simple recipe on a simple oven beat a complex recipe on a fancy oven.

B. Taking More Photos Doesn't Always Help (Shot Count)

• The Analogy: Imagine trying to guess the average height of people in a crowd by taking photos. If you take 1 photo, you might get a lucky guess. If you take 1,000 photos, you get a very accurate average. But if you take 10,000 photos, you aren't getting much more accuracy; you're just wasting film and time.
• The Result: Increasing the number of measurements (shots) helps a lot when you go from very few to a moderate amount (like 256 to 1,000). But once you hit that sweet spot, taking even more shots just costs you more money without making the cake taste much better.

C. The "Noise Canceling" Headphones (Error Mitigation)

• The Analogy: The oven makes a loud humming noise (errors). You can buy "noise-canceling headphones" (error mitigation) to fix the sound.
  • Level 1 (Basic Headphones): These work well. They clean up the noise and give you a clearer picture of the cake.
  • Level 2 (Super Headphones): These are expensive and require a lot of extra processing. Sometimes they work great, but often they just make the sound weirder or cost too much for the tiny improvement they offer.
• The Result: Basic noise cancellation (Level 1) is usually worth it. The super-expensive version (Level 2) is often a waste of money for small problems like this.

D. The "VIP Lounge" vs. The "Regular Line" (Session vs. Single Job)

• The Analogy:
  • Single Job: You walk up to the oven, put your cake in, wait for it to bake, take it out, and leave. You pay only for the time the oven was actually baking.
  • Session Mode: You buy a "VIP pass" that lets you reserve the oven for an hour. You can put cakes in and out as fast as you want without waiting in line.
• The Result: Many people thought the VIP pass (Session Mode) would give a better cake because the oven wouldn't "drift" or change its temperature between batches. They were wrong.
  • For small cakes, the oven doesn't change temperature that fast anyway.
  • The Catch: The VIP pass charges you for the entire hour you reserved, even if you only baked for 5 minutes. The regular line charges you only for the 5 minutes you baked.
  • Conclusion: For small tasks, the "Regular Line" (Single Job) is much cheaper and gives the exact same quality cake. Don't pay for the VIP pass unless you are baking a massive banquet.

3. The Bottom Line for Beginners

If you are new to quantum computing and want to run a calculation:

1. Keep it simple: Use the simplest recipe (circuit mapping) possible.
2. Don't overdo it: Don't run thousands of measurements if a few hundred are enough.
3. Skip the VIP pass: Just run your jobs one by one. It's cheaper and just as accurate for now.
4. Use basic noise cancellation: It helps, but don't buy the most expensive version unless you have to.

The Takeaway: You don't need to be a quantum wizard or use the most expensive settings to get good results. Sometimes, the "out-of-the-box" settings, if chosen wisely, are the most efficient way to get the job done. The paper provides a "menu" for new users so they don't accidentally order the most expensive meal that tastes the same as the cheap one.

Technical Summary

1. Problem Statement

The Variational Quantum Eigensolver (VQE) is a leading algorithm for extracting ground-state energies from Noisy Intermediate-Scale Quantum (NISQ) devices. However, practical application on real hardware is hindered by a lack of intuitive understanding regarding how standard workflow parameters translate into achievable accuracy, runtime, and billed costs.

• The Gap: Existing literature often focuses on algorithmic improvements or simulator-based studies, lacking systematically collected, hardware-executed reference data. Users, particularly non-specialists, struggle to balance trade-offs between shot counts, backend selection, circuit mappings, error mitigation, and execution modes (session vs. single-job).
• Objective: To provide a hardware-validated reference dataset and analysis for the hydrogen molecule ($H_2$) to establish a transparent baseline for assessing current IBM Quantum hardware capabilities and guiding user workflow choices.

2. Methodology

The study employs a standardized, "out-of-the-box" workflow using the Qiskit 2.x software stack (specifically Qiskit Nature and Qiskit Runtime) to minimize user intervention and reflect typical practitioner experiences.

• Benchmark System: The electronic ground state of the hydrogen molecule ($H_2$) at an internuclear distance of 0.735 Å. This system allows for an exact classical reference energy ($E_{ref} = -1.85727503$ a.u.) via NumPy diagonalization, isolating errors to the quantum workflow rather than the chemical model.
• Software Stack:
  • Tools: Qiskit Nature (0.7.2), Qiskit Algorithms (0.3.1), Qiskit Runtime (0.40.1).
  • Ansatz: Unitary Coupled Cluster (UCC) with a Hartree-Fock reference, one repetition, and real-valued parameters.
  • Optimizers: Primarily COBYLA (chosen for rapid convergence in low-dimensional spaces) with SPSA used for comparative reference.
• Variables Tested:
  • Mappings: Four fermion-to-qubit mappings were tested: Jordan-Wigner (JW, 4 qubits), Parity (P, 4 qubits), Parity with particle-number tapering (PF, 2 qubits), and Fully Tapered Parity (PT, 1 qubit).
  • Shot Counts: Ranging from 1 to 8192 (nominal), mapped to Estimator precision ($\epsilon = 1/\sqrt{N_{shots}}$).
  • Resilience Levels: 0 (no mitigation), 1 (measurement error mitigation), and 2 (includes zero-noise extrapolation/gate twirling).
  • Execution Modes: Session (exclusive backend access for a time window) vs. Single-job (independent job submission per iteration).
  • Backends: Experiments ran on multiple IBM Quantum processors (Heron r1/r2/r3, Eagle r3, Nighthawk r1) between August 2025 and April 2026.
• Metrics: Accuracy (absolute energy deviation $E_{err}$), Computational Effort (optimizer iterations, wall-clock time), and Economic Cost (billed time).

3. Key Contributions

1. Hardware-Validated Reference Dataset: A comprehensive collection of VQE results across diverse backends, shot budgets, and workflow configurations, providing a rare empirical baseline for $H_2$.
2. Workflow Trade-off Analysis: A systematic quantification of how specific choices (mapping, shots, mitigation, execution mode) impact the accuracy-cost curve.
3. Guidance for Non-Specialists: A set of evidence-based recommendations for users entering quantum chemistry, prioritizing transparency and cost-efficiency over aggressive, unproven optimizations.
4. Re-evaluation of Best Practices: Challenging common assumptions, such as the universal benefit of session-based execution or the linear scaling of accuracy with shot counts.

4. Key Results

A. Circuit Mapping and Complexity

• Dominant Factor: Circuit simplification via symmetry tapering is the most consistent driver of accuracy.
• Performance: The PT (1-qubit) and PF (2-qubit) mappings significantly outperformed the standard 4-qubit JW and P mappings.
  • PT reduced the problem to a single parameterized rotation, minimizing gate depth and error accumulation.
  • JW and P circuits (48 and 32 CNOTs respectively) suffered from higher gate error rates and decoherence, leading to larger energy deviations.
• Conclusion: Reducing circuit depth and qubit count via mapping is more effective for accuracy on current hardware than increasing shot counts or using complex error mitigation.

B. Shot Count and Accuracy

• Diminishing Returns: Increasing shot counts beyond the 256–1024 range yielded diminishing returns in accuracy.
• Noise Floor: For most backends, the energy error plateaued due to hardware noise (gate errors, readout errors) rather than sampling noise. Simply increasing shots could not overcome the systematic bias introduced by the hardware.
• Cost Scaling: Quantum execution time scales linearly with shot count. 1024 shots emerged as a practical compromise between stability and cost.

C. Error Mitigation (Resilience)

• Level 1 (Measurement Mitigation): Provided a robust and consistent improvement in accuracy across all backends, though at a significant cost premium (additional circuit executions).
• Level 2 (Advanced Mitigation): Results were variable. While it improved accuracy on some backends, it degraded results on others (e.g., ibm_aachen) and incurred the highest cost. This suggests that aggressive mitigation can amplify statistical noise or introduce bias when sampling is limited.

D. Execution Modes: Session vs. Single-Job

• Accuracy: Session-based execution did not provide a systematic accuracy advantage over single-job execution. The short duration of the VQE runs (minutes) meant that hardware calibration drift was negligible in both modes.
• Cost: Session execution incurred substantially higher billed time (minutes) compared to single-job execution (seconds), as sessions bill for the entire reserved time window, including idle time between iterations.
• Conclusion: For small-to-medium VQE workloads, single-job execution is significantly more cost-efficient without sacrificing accuracy.

E. Backend Variability

• Different backends exhibited distinct noise profiles and execution times. Newer architectures (e.g., Heron r2/r3) generally outperformed older ones (Heron r1), but variability within a single chip (due to qubit selection stochasticity) was also significant.

5. Significance and Recommendations

This study provides a critical reality check for the quantum chemistry community, shifting the focus from "how to run VQE" to "how to run VQE efficiently."

• For Practitioners:
  • Prioritize Simplicity: Use circuit tapering (PT/PF) to reduce qubit count and depth before attempting complex error mitigation.
  • Default to Single-Job: Unless the workflow is extremely long (hours), single-job execution is the most cost-effective strategy.
  • Moderate Shots: Do not blindly increase shot counts; 1024 is often sufficient for small molecules.
  • Selective Mitigation: Use Resilience Level 1 if cost permits, but avoid Level 2 unless specific hardware characteristics justify it.
• For the Field: The paper establishes that for current NISQ devices, circuit complexity is the primary bottleneck for accuracy, not just sampling noise. It highlights that "out-of-the-box" workflows, when properly mapped, can yield reliable results without bespoke engineering, provided users understand the inherent cost-accuracy trade-offs.

The authors conclude that while default workflows are effective for small problems, scalability to larger molecules will require re-evaluating these trade-offs, potentially necessitating more sophisticated scheduling or error mitigation strategies in the future.