Simulating Quantum Computers: Innovation in Verification

By Bernard Murphy | December 29, 2025 | 6:00 AM
Categories: Cadence, EDA, Quantum Computing

---

Introduction

Quantum computing is transitioning from theoretical research to practical engineering reality. As investment and innovation accelerate, a fundamental challenge emerges: how do we verify quantum algorithms and circuits before reliable, large-scale quantum hardware is widely available?

The answer lies in quantum simulation on classical computers. While verification through simulation is familiar in classical digital design, quantum simulation introduces fundamentally different mathematical models, computational scaling challenges, and verification philosophies. This blog explores recent research that sheds light on how quantum circuits can be simulated, understood, and validated using classical computing infrastructure.

---

Why Quantum Simulation Matters

Quantum algorithms must be validated long before they can be deployed on physical quantum machines. Unlike classical logic, quantum computation operates on entangled qubits, probabilistic states, and complex-valued amplitudes. This makes direct reasoning about correctness extremely challenging.

Simulation becomes the first—and most essential—step in quantum verification:

• It enables validation of algorithm behavior

• It provides visibility into quantum state evolution

• It allows debugging without collapsing quantum states

• It builds foundational understanding for future verification methodologies

---

The Research Highlight

This month’s featured work is “How to Write a Simulator for Quantum Circuits from Scratch: A Tutorial”, published on arXiv in June 2025 by researchers from École de Technologie Supérieure (Montreal) and the University of Massachusetts.

The paper presents a hands-on, algorithmic tutorial for building a quantum circuit simulator capable of handling approximately 20 qubits. It avoids deep quantum physics and instead focuses on the computational and mathematical foundations required to simulate quantum algorithms on classical machines.

Key characteristics of the work include:

• Clear introduction to quantum notation and linear algebra

• Step-by-step construction of a quantum simulator

• Web-based implementations and code walkthroughs

• Focus on algorithmic efficiency rather than physical implementation

---

How Quantum Simulation Differs from Classical Simulation

In classical digital simulation:

• Each bit or register is independent

• Events are processed locally and sequentially

• Complexity scales linearly with circuit size

Quantum simulation is fundamentally different.

State Vector Explosion

A quantum system with n qubits has a state vector of size 2ⁿ, representing all possible combinations of qubit states simultaneously. Simulation involves:

• Maintaining a global state vector

• Applying quantum gates as matrix operations

• Updating all amplitudes in parallel

Naively, this requires O(4ⁿ) operations, which becomes infeasible very quickly.

---

Algorithmic Innovations

The featured paper introduces optimization techniques that reduce simulation complexity to approximately O(n · 2ⁿ) by exploiting properties of quantum gates and tensor operations. Instead of forming full matrices, gate operations are applied selectively to relevant parts of the state vector.

This makes simulation of small quantum systems practical on classical hardware and enables:

• Efficient handling of common gates (Hadamard, Pauli, SWAP, Toffoli)

• Qubit-wise operations without explicit matrix construction

• Direct manipulation of state indices for SWAP operations

---

GPU Acceleration and Practical Limits

Because quantum state vectors consist of floating-point complex numbers, quantum simulation maps naturally to GPU architectures. Recent demonstrations show:

• Simulation of up to 36 qubits using multiple high-end GPUs

• Significant speedups over CPU-only approaches

That said, exponential scaling remains unavoidable. Even with aggressive optimization, classical simulation currently caps at around 50 qubits on high-performance computing platforms.

---

Understanding Quantum Behavior Through Simulation

Each element of the quantum state vector represents a complex amplitude. The square of its magnitude corresponds to the probability of observing that state upon measurement. The entire state vector can be visualized as a point on the surface of a 2ⁿ-dimensional hypersphere.

Quantum gates perform rotations of this state vector, not Boolean logic. Designing a quantum algorithm is therefore an exercise in identifying the right sequence of rotations that move the system toward the desired result with high probability.

Simulation provides critical insight into this process—something physical quantum hardware cannot expose directly due to state collapse during measurement.

---

Perspectives from Verification Leaders

Paul Cunningham’s View

From a verification standpoint, quantum simulation resembles analog circuit simulation more than digital logic simulation. Every operation impacts the entire system state, and verification requires global reasoning rather than localized event handling.

This shift challenges decades of digital verification methodology and underscores the need for new tools, abstractions, and mental models.

---

Raúl Camposano’s View

The tutorial stands out for its accessibility. With roughly 1,000–2,000 lines of JavaScript, the simulator demonstrates how far careful algorithmic design can go, even on consumer hardware.

More importantly, the exercise is not about replacing commercial tools, but about deep understanding—a prerequisite for developing future verification and validation frameworks for quantum systems.

---

Looking Ahead: Verification Beyond Simulation

Simulation is verification step one—but it is not the endgame. As quantum algorithms grow in size and complexity, the industry will need:

• Formal reasoning methods for quantum correctness

• Abstractions that scale beyond brute-force simulation

• New verification paradigms inspired by both classical EDA and physics

Understanding simulation today lays the groundwork for verification innovation tomorrow.

---

Conclusion

Quantum computing challenges nearly every assumption embedded in classical design and verification. Simulation, though computationally expensive, remains the most powerful tool for validating quantum algorithms and building foundational insight.

As research progresses and tooling matures, the lessons learned from quantum simulation will directly influence the next generation of verification methodologies—ensuring that quantum innovation is not only powerful, but also correct.