In 2021, Renou et al published the paper “Quantum theory based on real numbers can be experimentally falsified” in Nature. It caused a decent sized splash at the time. A quick search revealed articles in Quanta and Scientific American and APS News, as well as a talk at QIP 2022.
In this post, I’m going to explain why the title of the paper is wrong.
The Goal
Before we get going, I need to warn you away from a common misunderstanding of Renou et al’s claim. Renou et al are NOT claiming that it’s impossible to write down math that behaves like quantum mechanics without using the word “complex” or the symbol “$\mathbb{C}$”. That would be vacuously wrong, because you can just mimic the behavior of complex numbers using pairs of real numbers (and appropriately tweaked definitions of operations). No one is claiming it’s impossible to write the C code struct complex { float real; float imag; }; . What Renou et al are actually claiming is that if you start with quantum mechanics, and then remove all operations and states involving non-real numbers, and then try to emulate what was lost using what remains, you will fail in an experimentally detectable way.
The way I like to think about this is in terms of quantum gatesets. The following gateset is known to be universal: the Toffoli gate (“CCX”), the Hadamard gate (“H”), the Measurement gate (“M”), and the 45° phasing gate (“T”). Roughly speaking: the Toffoli provides classical computation, the Hadamard provides superposition, the measurement provides results, and the T gate provides the complex plane. Only the T gate needs complex numbers to describe; the other three can be described using real numbers. What Renou et al are trying to do is make a test that this CHMT gateset (CCX + H + M + T) can pass, but the T-missing CHM gateset (CCX + H + M) can’t pass.
Imagine you’re buying a quantum computer from Eve. Eve claims the computer supports the CHMT gateset. You’re suspicious. You suspect it can’t actually do T gates, meaning it actually only supports the CHM gateset. Without T gates, you’d be stuck in the “real-only” subset of quantum computations. Is there some test you can do, some challenge you can give to Eve, that she can only pass if the T gate is working? Some kind of externally verifiable proof that the computer isn’t stuck in the real-only subset?
(A trivial solution would be to bring a trusted quantum computer and ask Eve to produce 100 copies of the state $|0\rangle + i|1\rangle$ and transmit them to the trusted computer. The trusted computer then simply verifies they all produce +1 when measured in the Y basis. But this solution doesn’t translate back to the physics use case that Renou et al have in mind, where basically the whole universe isn’t trusted, so I’m not going to allow it. Assume the verifier only has a classical computer.)
At first, you might think it’s easy to tell if the T gate is broken. Just ask Eve to do a quantum computation involving complex numbers. For example, Shor’s algorithm ends with a Quantum Fourier Transform which is defined in terms of complex values. Just ask Eve to factor some numbers. Alas, this doesn’t work. Although the gateset CHM isn’t strictly universal, it’s still computationally universal. Any CHMT circuit can be efficiently compiled into, and emulated by, the CHM gateset.
Hearing that, you may now think it’s obviously impossible to determine if Eve’s computer can’t do T gates. However, there’s still hope: the compilation into CHM may not be locality preserving. Operations that used to be local may now need to interact. This is how “Bell tests” distinguish the classical gateset CMT from the universal gateset CHMT. Classical computers can simulate quantum computers with exponential overhead, so a fast enough classical computer can technically pass any computational test you give. But, crucially, the classical simulation of quantum circuits isn’t locality-preserving. And there’s provably no way to fix that. So all you need to do is ask Eve to distribute the computation over multiple computers, with crucial steps done under spacelike-separation. There are tests, such as the CHSH test, that CHMT computers can pass when spacelike-separated but that CMT computers cannot pass when spacelike-separated.
What Renou et al attempt to do is come up with an analogue of the CHSH test: use locality to distinguish CHM vs CHMT. At this point in the post, it would perhaps be appropriate to describe the details of Renou et al’s test. Roughly speaking, they define a computation distributed over three quantum computers (Alice and Bob and Charlie) where Alice and Charlie are doing a generalized version of the CHSH inequality mediated by entanglement swapped through Bob. However, ultimately these details don’t matter. There are locality-preserving ways to compile CHMT circuits into CHM circuits, making the whole idea moot.
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