⚛️ quantum physics

Kardashev scale Quantum Computing for Bitcoin Mining

This paper demonstrates that while quantum computers theoretically offer a quadratic speedup for Bitcoin mining via Grover's algorithm, the immense physical resource requirements—ranging from national-grid-scale energy for partial preimage attacks to Kardashev Type II energy levels for full mainnet difficulty—render practical quantum mining infeasible for any foreseeable civilization.

Original authors: Pierre-Luc Dallaire-Demers

Published 2026-03-27

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Pierre-Luc Dallaire-Demers

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Question: Can a Quantum Computer Steal Bitcoin by Mining?

Everyone is worried that quantum computers will break Bitcoin. Most people are worried about Shor's Algorithm, which could steal your coins by cracking your digital signature (like picking the lock on your front door).

But this paper asks a different question: Could a quantum computer become so good at "mining" (solving the math puzzles to create new blocks) that it takes over the entire Bitcoin network?

The short answer from the paper is: No. Not even close.

To understand why, we have to look at the difference between "theoretical speed" and "physical reality."

1. The Theoretical Speedup: The "Magic Magnifying Glass"

Imagine Bitcoin mining is like looking for a specific grain of sand on all the beaches on Earth.

Classical Computers (ASICs): These are like millions of people with shovels, digging one grain at a time. It takes a long time, but they are very efficient at shoveling.
Quantum Computers (Grover's Algorithm): Theoretically, a quantum computer has a "magic magnifying glass." Instead of checking every grain, it can check them in a way that finds the right one in the square root of the time.

If it takes a classical computer 100 years to find the grain, a quantum computer might only take 10 years. That sounds like a huge win, right?

The Catch: The paper argues that while the math says you save time, the machinery required to build that magnifying glass is so massive and energy-hungry that it cancels out the benefit.

2. The Three "Tax" Bills

The paper breaks down why the quantum miner fails. It's not just about the search; it's about the overhead. Think of it like trying to drive a race car, but you have to pay three massive taxes before you can even start the engine.

Tax #1: The "Reversible" Engine (The Oracle)

To use the quantum magnifying glass, you have to build a reversible engine. In classical computing, you can erase your notes as you go. In quantum computing, you can't erase anything; you have to "un-do" every step perfectly to keep the quantum state alive.

The Analogy: Imagine you are solving a maze. A classical person draws a line, hits a wall, erases the line, and tries again. A quantum person has to draw the line, hit the wall, and then magically un-draw the line without leaving a single scratch on the paper, all while the paper is vibrating.
The Cost: This "un-drawing" process requires a massive amount of extra computing power (gates), inflating the cost of a single search by a factor of roughly 200 billion.

Tax #2: The "Error Correction" Tax (The Distillation Factory)

Quantum bits (qubits) are incredibly fragile. They are like glass marbles; a tiny vibration breaks them. To keep them stable, you need "error correction."

The Analogy: To keep one "logical" qubit (the smart one doing the work) safe, you need thousands of "physical" qubits (the bodyguards) constantly checking on it.
The Cost: The paper calculates that for every single math operation the quantum computer does, it needs to run a "magic factory" to produce special states to fix errors. This factory takes up thousands of times more space than the actual computer.

Tax #3: The "Fleet" Tax (The Time Limit)

Bitcoin has a strict rule: a new block must be found every 10 minutes.

The Analogy: Imagine a race where you have 10 minutes to find a needle in a haystack. Even with your magic magnifying glass, if your machine is too slow to finish the job in 10 minutes, you lose.
The Cost: Because the quantum machine is so slow (due to the two taxes above), you can't just build one big machine. You have to build billions of them and run them all at the same time to have a chance of winning within 10 minutes.

3. The Kardashev Scale: How Big is "Too Big"?

The authors take these numbers and plug them into the Kardashev Scale, which measures how advanced a civilization is based on how much energy it consumes.

Type I: A civilization that uses all the energy of its planet (Earth).
Type II: A civilization that uses all the energy of its star (The Sun).
Type III: A civilization that uses all the energy of its galaxy.

The Result:

To mine Bitcoin with a quantum computer at today's difficulty, you would need a fleet of computers consuming 10^25 Watts of power.
The Comparison: The entire Bitcoin network currently uses about 15 Gigawatts (roughly the power of a small country).
The quantum fleet would need 100 trillion times more power than the current Bitcoin network.
The Verdict: This energy requirement is close to the power output of the Sun (Type II Civilization).

4. The "Science Fiction" Escape Hatch

The paper even asks: "What if we build a quantum computer using super-fast, high-energy physics (like nuclear or Planck-scale physics)?"

They imagine a computer that runs at the speed of light or uses nuclear forces.

The Result: Even if you build a computer that runs a billion times faster, the energy required to keep it cool and stable becomes so high that it turns the computer into a star.
The Metaphor: It's like trying to power a lightbulb by building a nuclear reactor inside the bulb. You get the light, but you've also created a supernova. The machine stops looking like a computer and starts looking like an engineered astrophysical object.

Summary: The Final Verdict

The paper concludes that while quantum computers are scary for stealing Bitcoin (breaking signatures), they are completely useless for mining Bitcoin.

The Myth: "Quantum computers will be so fast they will mine all the Bitcoin."
The Reality: "Quantum computers are so fragile and energy-hungry that building one big enough to mine Bitcoin would require more energy than the Sun produces."

In everyday terms:
Trying to use a quantum computer to mine Bitcoin is like trying to win a footrace by building a rocket ship. Sure, the rocket is faster, but the fuel required to launch it costs more than the prize money, and the exhaust would blow up the track. The classical miners (the people with shovels) are still the most efficient way to do the job.

The takeaway for Bitcoin: The "mining" part of Bitcoin is safe from quantum computers for the foreseeable future. The only real danger is the "signature" part (your wallet), which is a different problem entirely.

1. Problem Statement

The paper addresses a critical ambiguity in the discourse surrounding Bitcoin and quantum computing. While it is widely accepted that Shor's algorithm threatens Bitcoin's digital signatures (ECDSA) in the near future, the threat to Bitcoin mining (Proof-of-Work) via Grover's algorithm is often misunderstood.

Public discussion frequently conflates the asymptotic quadratic speedup of Grover's algorithm ( $O(\sqrt{N})$ ) with practical feasibility. The paper argues that while Grover's algorithm theoretically halves the exponent of the search space, the physical resource costs required to implement a fault-tolerant Grover miner for Bitcoin are astronomical. Previous studies often stopped at asymptotic query counts or buried the physical costs in complex surface-code ledgers, failing to provide an end-to-end cost analysis that includes reversible oracle construction, magic-state distillation, fleet logistics, and energy consumption.

2. Methodology

The authors developed an open-source, end-to-end estimator that sweeps the full attack surface of quantum Bitcoin mining. The methodology proceeds through four distinct layers:

Reversible Oracle Construction:
- The paper models the Bitcoin mining oracle as a reversible double-SHA-256 function (for block headers) and a RIPEMD-160(SHA-256(pubkey)) function (for P2PKH address preimages).
- Using 32-bit Cuccaro–Draper–Kutin–Moulton (CDKM) ripple adders and Gidney's measurement-assisted ANDs, they calculated the exact logical gate counts.
- Key Finding: A single Grover iteration requires $\sim 3 \times 10^5$ non-Clifford (T) gates, inflating the naive query count by a factor of $\sim 2^{38}$ .
Fault-Tolerance & Surface Code Mapping:
- The logical circuits are mapped to the surface code, the leading candidate for fault-tolerant quantum computing.
- The model accounts for magic-state distillation, which is required for T-gates. A single "factory" occupies thousands of physical qubits to produce high-fidelity T-states.
- The authors calculate the necessary code distance ( $d$ ) based on a logical failure budget, determining the total physical qubit count ( $Q_{machine}$ ) required for a single mining machine.
Fleet Logistics (Nakamoto Consensus Constraints):
- Bitcoin mining is time-constrained (10-minute block window). A single quantum machine cannot run enough Grover iterations within this window to achieve a high probability of success.
- The model calculates the required fleet size ( $N_{machines}$ ) by running many independent machines in parallel to meet a target success probability ( $P_t$ ) within the time cap ( $t_{cap}$ ).
- This introduces an exponential scaling factor: as difficulty increases, the number of required machines grows exponentially to compensate for the limited runtime per machine.
Energy & Kardashev Scale Analysis:
- The total physical qubit count is converted into wall-plug power using conservative estimates for different architectures (Superconducting, Neutral Atom, Ion Trap).
- The resulting power demands are mapped against the Kardashev scale (Type I: Planetary, Type II: Stellar, Type III: Galactic) to contextualize the energy requirements.

3. Key Contributions

First End-to-End Costing: This is the first study to total the physical costs of quantum Bitcoin mining, moving from abstract query counts to concrete hardware requirements (qubits, factories, power).
The "Three-Compounding-Overhead" Model: The paper identifies three specific factors that destroy the practicality of Grover mining:
1. Oracle Overhead: The massive T-gate count of reversible SHA-256.
2. Distillation Tax: The huge physical footprint of magic-state factories required to supply T-gates.
3. Fleet Multiplication: The exponential number of machines needed to satisfy the 10-minute block deadline.
Open-Source Estimator: The authors released a tool (kardashev repository) allowing the community to sweep difficulty bits, runtime caps, and success probabilities across different hardware architectures.
High-Energy Surface Code Hypothesis (HESC): The paper explores speculative "science-fiction" scenarios where surface codes operate at higher energy scales (e.g., nuclear or Planck scales) to increase clock speeds. Even under these extreme assumptions, the resource requirements remain prohibitive.

4. Key Results

The results demonstrate a sharp transition where quantum mining becomes physically impossible for any civilization below a Kardashev Type II level.

Partial Preimage (Favorable Case):
- Setting: Difficulty $b=32$ (approx. 2224 marked states).
- Result: A superconducting fleet requires $\sim 10^8$ physical qubits and $\sim 10^4$ MW (10,000 MW).
- Context: This power load is comparable to a large national grid.
Mainnet Difficulty (Realistic Case):
- Setting: Bitcoin mainnet difficulty as of Jan 2025 ( $b \approx 79$ ).
- Result: The bill explodes to $\sim 10^{23}$ physical qubits and $\sim 10^{25}$ W.
- Context: This power demand approaches the Kardashev Type II threshold (stellar energy output, $\sim 10^{26}$ W).
True Preimage (Worst Case):
- Setting: Finding a specific hash with 1 marked state ( $b=256$ ).
- Result: Requires $\sim 10^{75}$ qubits, exceeding any conceivable astrophysical power source.
Quantum-to-Classical Ratio:
- At current mainnet difficulty, the quantum fleet requires roughly $10^{14}$ times more power than the classical ASIC network. Even in the most favorable partial-preimage scenario, the quantum miner is $10^{10}$ times less energy-efficient.
Impact of Runtime Caps:
- Reducing the runtime cap (e.g., from 10 minutes to 60 seconds) forces an exponential increase in fleet size, shifting the requirements from "industrial scale" to "civilization scale" almost instantly.

5. Significance and Conclusion

The paper fundamentally shifts the narrative regarding Bitcoin's quantum security:

Mining is Safe (for now and the foreseeable future): Unlike the signature layer, which is vulnerable to Shor's algorithm, the mining layer is not a practical threat from quantum computers. The quadratic speedup of Grover's algorithm is completely negated by the multiplicative overheads of fault tolerance, magic-state distillation, and the 10-minute consensus deadline.
Physical Impossibility: To mount a successful quantum mining attack that impacts Nakamoto consensus, an adversary would need to construct a machine with the energy output of a star (Type II civilization). This is far beyond the capabilities of any near-term or mid-term technological civilization.
Policy Implications: The paper suggests that while the signature layer requires urgent migration to post-quantum cryptography, the mining layer does not require immediate protocol changes due to quantum threats. The "quantum mining" threat is a theoretical curiosity rather than a practical engineering challenge.

In summary, the paper concludes that practical quantum mining collapses under its own physical weight. The "Grover advantage" is an asymptotic illusion that vanishes when translated into the physical reality of error-corrected, energy-constrained hardware.