Quantum Resource Estimation — Comparative Study

A systematic quantum resource estimation (QRE) study across four algorithm families using the Microsoft Azure Quantum Resource Estimator. Circuits are implemented in Q#, with resource analysis across two physical qubit models.

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Algorithms Studied

Algorithm	Tool	Sizes
Grover's Search	Q#	5, 10, 15, 20, 25 qubits
Heisenberg XXX Hamiltonian Simulation	Q#	4, 8, 12, 16, 20 qubits
Quantum Phase Estimation (QPE)	Q#	4, 6, 8, 10, 12 counting qubits
VQE Ground State Estimation (H₂, LiH, BeH₂)	Q#	4, 12, 14 qubits

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Key Findings

• T-factory distillation consumes 75–98% of all physical qubits across every algorithm studied
• Improving physical error rate from 10⁻³ → 10⁻⁴ reduces physical qubit cost by 5–17×
• Logical resources are hardware-independent — physical resources are entirely driven by QEC assumptions
• VQE is the most resource-efficient algorithm for near-term fault-tolerant hardware
• Grover's search shows the steepest resource scaling — up to 752,318 physical qubits at 25 qubits

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Project Structure

quantum-resource-estimation/
│
├── qsharp/                         # Q# circuit implementations
│   ├── Grover.qs                   # Grover search (5q–25q)
│   ├── Heisenberg.qs               # Heisenberg XXX simulation (4q–20q)
│   ├── QPE.qs                      # Quantum phase estimation (4–12 counting)
│   └── Chemistry.qs                # VQE ansatz (H₂, LiH, BeH₂)
│
├── results/                        # Azure RE output screenshots
│   ├── grover_re.png
│   ├── heisenberg_re.png
│   ├── qpe_re.png
│   └── chemistry_re.png
│
├── report/
│   └── azure_re_report.md          # Full analysis report
│
└── README.md

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Hardware Configurations

All circuits estimated under two physical qubit models using surface code QEC and error budget = 0.001:

Parameter	qubit_gate_ns_e3	qubit_gate_ns_e4
Gate time	1 ns	1 ns
Physical error rate	10⁻³	10⁻⁴
QEC scheme	Surface code	Surface code

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How to Run

1. Install the QDK extension in VS Code
2. Open any .qs file in qsharp/
3. Move @EntryPoint() to the circuit size you want
4. Press Ctrl+Shift+P → Q#: Calculate Resource Estimates
5. Select qubit_gate_ns_e3 and/or qubit_gate_ns_e4

No installation or Python environment required.

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Results Summary

Physical Qubits Required (e3 / e4)

Algorithm	Min	Max (e3)	Max (e4)
Grover	19,400	752,318	77,598
Heisenberg XXX	14,470	1,041,210	105,252
QPE	22,900	513,084	76,244
VQE Chemistry	13,750	232,320	25,000

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Tools & Stack

Component	Technology
Circuit implementation	Q# (Microsoft QDK)
Resource estimation	Azure Quantum Resource Estimator
QEC model	Surface code
Development environment	VS Code Web (vscode.dev/quantum)

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References

• Azure Quantum Resource Estimator
• Microsoft QDK Documentation
• Qiskit Documentation
• Babbush et al., "Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity" (2018)
• Bauer et al., "Quantum Algorithms for Quantum Chemistry and Quantum Biology" (2020)