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Computational speed

The compute speed difference between quantum computing (QC) and High-Performance Computing (HPC) is not a fixed number, but a fundamental difference in scaling that results in an exponential speedup for a specific class of problems.

The speed difference is generally qualitative (exponential) rather than quantitative (a fixed factor), and it only applies when a suitable quantum algorithm exists for a problem that is computationally hard for classical computers.

Qualitative Speed Difference: Exponential vs. Polynomial

The primary speed distinction lies in how the time required to solve a problem scale as the problem size increases:

• Quantum Computing (QC): For certain problems, quantum algorithms – like Shor’s algorithm for factoring, or the Quantum Echoes algorithm for physics simulation – offer a super-polynomial or exponential speedup over the best-known classical algorithms.
  • This means that as a problem’s size grows (e.g., more variables), the time it takes the quantum computer to solve it grows much more slowly than the time it takes a classical computer. The performance gap keeps growing exponentially.
• High-Performance Computing (HPC): Classical HPC systems typically experience a polynomial increase in computation time as problem size increases, but for certain “hard” problems (like factoring large numbers), the required time scales exponentially for classical machines.

Quantitative Example (Specific Problem)

The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. For instance, the use of a processor with programmable superconducting qubits to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 253 (about 1016). Measurements from repeated experiments sample the resulting probability distribution, with a verification by using classical simulations. The Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times—our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy for this specific computational task, heralding a much-anticipated computing paradigm (2019 October, Nature).

Up to 43 qubits, it is used by a Schrödinger algorithm, which simulates the evolution of the full quantum state; the Jülich supercomputer (with 100,000 cores, 250 terabytes) runs the largest cases. Above this size, there is not enough random access memory (RAM) to store the quantum state. For larger qubit numbers, we use a hybrid Schrödinger–Feynman algorithm running on Google data centres to compute the amplitudes of individual bitstrings. This algorithm breaks the circuit up into two patches of qubits and efficiently simulates each patch using a Schrödinger method, before connecting them using an approach reminiscent of the Feynman path-integral. Although it is more memory-efficient, the Schrödinger–Feynman algorithm becomes exponentially more computationally expensive with increasing circuit depth owing to the exponential growth of paths with the number of gates connecting the patches.

To estimate the classical computational cost of the supremacy circuits, we ran portions of the quantum circuit simulation on both the Summit supercomputer as well as on Google clusters and extrapolated to the full cost.

Context and Limitations

It is important to note the following:

• Hybrid Systems: The articles you provided emphasize that the immediate future of computing involves hybrid quantum-classical systems, where quantum computers act as specialized accelerators for the most difficult computational kernels that would be intractable for the classical HPC system alone.
• Not Universal: Quantum computers are not faster at all problems. Classical computers are often faster for small problem sizes and for tasks that do not benefit from quantum mechanical principles.
• Slower Steps: A single quantum computational step is typically much slower than a classical one. The speedup comes from the ability of quantum mechanics (superposition, entanglement, and interference) to solve the problem in far fewer steps overall.