Why Quantum Computers Cannot Try Every Answer

TL;DR

Quantum computers do not solve hard search problems by testing every answer at once, because measuring an equal superposition produces only a random result. A useful quantum algorithm must instead arrange interference so amplitudes cancel for wrong answers and reinforce for the right answer, while quantum error correction makes scalability a staggeringly difficult engineering challenge rather than requiring perfectly isolated qubits.

Transcript

So today we have Scott Aaronson, a UT professor of CS, and also a blogger at Shtetl-Optimized. And on the top of your blog, you have something that says, "If you take just one piece of information away from this blog, quantum computers would not solve hard search problems instantaneously by simply trying all the possible solutions at once." Why not... Read More

Key Insights

• Quantum computers cannot obtain every candidate solution merely by placing qubits into a superposition, because measurement converts the state into a single outcome. Without additional interference operations, an equal superposition yields a random answer rather than the correct answer to a search or optimization problem.
• A quantum amplitude is a number assigned to each possible measurable state of a physical system. Larger amplitudes correspond to more likely outcomes after applying the measurement rule, but amplitudes differ from probabilities because they may be negative or even complex numbers.
• Quantum interference works through amplitudes that can reinforce or cancel one another. If two paths to an outcome contribute positive and negative amplitudes, they can cancel completely, while contributions with compatible amplitudes can increase the chance that an outcome appears during measurement.
• A system containing 100 qubits requires two to the hundredth power amplitudes to describe its quantum state. It is easy to create an equal superposition over all configurations, including every possible cryptographic key or candidate solution, but that alone provides no useful computational answer.
• The probability of observing a particular quantum outcome is the squared absolute value of its amplitude. Consequently, measuring an equal superposition without first transforming its amplitudes produces a uniformly random candidate, something that could be generated much more simply by flipping coins.
• The key to quantum speed advantage is choreographing interference without already knowing the correct answer. A successful algorithm must make paths toward wrong answers cancel while making contributions toward the right answer reinforce, which is possible only for computational problems with suitable structure.
• Quantum computing is both a technology project and a fundamental scientific test. It applies quantum mechanics as formulated in 1926 to the previously untested regime of universal quantum computation and quantum error correction, while practical discussions often focus prematurely on near-term industry applications.
• Quantum error correction and fault tolerance changed expert views about scalability during the 1990s. Their central implication is that physical qubits need not be perfectly isolated from their environment, but they must still be isolated extraordinarily well, leaving a staggeringly difficult engineering problem.

Questions & Answers

Q: Why can quantum computers not try every answer at once?

Quantum computers can represent an equal superposition containing amplitudes for every possible answer, but representation is not the same as retrieving all those answers. When the state is measured directly, quantum mechanics returns only one outcome, with probability determined by the squared absolute value of its amplitude. An unchanged equal superposition therefore supplies a random answer, not an immediate solution to a hard search problem.

Q: What is a quantum amplitude and how does it differ from probability?

A quantum amplitude is a number assigned to each possible measurable state of a physical system. Its squared absolute value determines the probability of observing that state. Unlike an ordinary probability, an amplitude can be negative or complex. This difference allows separate computational paths to reinforce or cancel one another, producing interference that ordinary positive probabilities cannot reproduce.

Q: How does interference make a quantum computer useful?

Interference lets a quantum algorithm reshape the probabilities of its possible outputs before measurement. The intended pattern makes positive and negative contributions toward wrong answers cancel through destructive interference, while contributions toward the correct answer reinforce one another. The difficult part is designing operations that create this pattern without knowing the correct answer in advance, so superposition alone is not enough.

Q: What happens when an equal quantum superposition is measured?

Measuring an equal superposition without performing any further useful operations returns a random configuration. Every candidate begins with an equal amplitude, and the measurement probability for each outcome is the squared absolute value of that amplitude. The result is therefore no better than generating a random answer by flipping coins, despite the state having contained amplitudes for all candidates before measurement.

Q: How many amplitudes are needed to describe 100 qubits?

A quantum state of 100 qubits requires two to the hundredth power amplitudes, with an amplitude associated with every possible bit configuration. A quantum computer can create an equal superposition across those configurations relatively easily. However, the enormous number of amplitudes does not allow every value to be read out, because measurement produces only one result according to the quantum probability rule.

Q: Why is quantum computing considered fundamental science?

Quantum computing tests quantum mechanics in the new regime of universal quantum computation and quantum error correction. The project does not require replacing the known laws of physics, since it takes the mathematical rules written down in 1926 seriously and attempts to realize their implications at computational scale. Establishing that nature supports such machines is therefore a scientific objective before questions about commercial applications.

Q: What did quantum error correction change about quantum computing?

Quantum error correction and quantum fault tolerance changed how many experts viewed scalable quantum computing during the 1990s. They showed that a scalable machine would not require qubits to be perfectly isolated from their environment, which would be physically absurd. Instead, qubits must be isolated extraordinarily well and protected through correction methods, reducing the obstacle to a staggeringly hard engineering problem.

Q: Why are quantum speedups difficult even with perfect hardware?

Perfect quantum hardware would remove engineering issues such as physical errors, but it would not automatically provide a computational speed advantage. An algorithm would still need to choreograph amplitudes so paths leading to wrong answers cancel and paths leading to the right answer reinforce. Finding this interference pattern without knowing the answer beforehand is the conceptual challenge that determines which problems the quantum approach can help solve.

Summary & Key Takeaways

• A quantum state assigns an amplitude to every possible configuration, and 100 qubits require two to the hundredth power amplitudes. Unlike ordinary probabilities, amplitudes may be negative or complex. Their ability to reinforce or cancel through interference accounts for the unusual behavior that quantum algorithms seek to exploit computationally.

• Creating an equal superposition across every candidate answer is easy, but measuring it directly returns a random answer because outcome probabilities equal the squared absolute values of their amplitudes. Quantum speedups therefore require algorithms that suppress wrong outcomes through destructive interference and strengthen the desired outcome through reinforcing contributions.

• Quantum computing remains fundamental science as well as a proposed technology because universal quantum computation and quantum error correction test established quantum mechanics in a regime not previously realized. Quantum error correction changed expert expectations by showing that scalable machines need extremely well-isolated qubits, rather than physically impossible, perfectly isolated ones.