# Unexpected Problems For Quantum Money

Proving that quantum money can never be copied–even by the banks that created it–turns out to be harder than expected.

In 1969, Stephen Wiesner at Columbia University suggested that the quantum properties of photons could be used to make “quantum money” that was impossible to counterfeit. The idea was to store a few dozen photons in light traps in each bill. and ensure that the polarisation of these photons was known only to the bank.

Since quantum states are impossible to copy, such a bank note could never be copied. And anyone wanting to check the note would need only take it to the issuing bank which could use its prior knowledge of the polarisations to test the bill’s veracity.

Wiesner’s idea became an inspiration for the generation of quantum physicists who developed quantum encryption, the ability to send a message with perfect security.

But there’s a practical problem with Wiesner’s quantum money. The most serious drawback is that only the issuing bank can verify that a bill is genuine whereas one of the important features of any practical currency is that anybody must be able to determine its veracity.

What’s needed is some kind of asymmetric technique that allows a bank to create quantum money that cannot be copied but also allows anyone to check it.

As it happens, something very similar is possible with so-called public key encryption techniques. Here anybody can encode a message with a publicly available key but the encrypted message can only be decoded with another key which is kept private.

Public-key encryption depends on certain types of mathematical functions that are easy to calculate in one direction but hard to do in reverse. The most famous example is multiplication. It’s easy to multiply two numbers together to get a third. But the problem of starting with the third number and working out which two generated it, a process called factoring, is much harder.

The security of public key encryption techniques rests on the idea that factoring can always be made so hard that it is effectively impossible for any conventional computer to do it; that is any computer that relies only on classical mechanics to do its number crunching.

Is it possible to design similarly asymmetric protocols that make quantum money possible?

One idea is to have the bank to write down a description of a quantum state that can be efficiently generated and then to manufacture the state in that. Of course, this description must be kept secret. The bank then constructs an algorithm to verify the state (but not reproduce it), a so-called verification circuit.

Quantum money then consists of both the quantum state and the verification circuit. Of course, if anybody can work out the secret description, they can print as many copies of the quantum money they like. But the security of quantum money relies on the difficulty of deducing the secret description given both the verification circuit and a copy of the state, which the money contains.

But there’s a problem. The bank knows the secret description and so can make as many copies as it likes of this money without anybody being any the wiser.

Today Andrew Lutomirski and a crack team of quantum eggheads at the Massachusetts Institute of Technology in Cambridge suggest how to close this loophole with an entirely new kind of quantum money which they call collision free.

Their idea is to use an entirely different state for the quantum money. This state is a superposition of an exponentially large number of unrelated terms each of which is created by the measurement of an equally exponential superposition. Incorporating this quantum measurement into the process of creating the quantum money ensures that a bank cannot reproduce this state, even though it knows how the initial superposition was created. At least, the bank cannot do this in any reasonable amount of time.

Lutomirski say that this form of quantum money can be verified using a Markov chain algorithm.

That’s an interesting development but the MIT team’s paper has a sting in the tail. Lutomirski and co say they expect that computationally secure collision-free quantum money is possible but are unable to give a proof.

“Surprisingly, the question of whether public-key quantum money schemes are possible under computational assumptions has remained open for forty years, from Wiesner’s time until today.”

And they finish with this jaw dropper: “Much as we wish it were otherwise, it seems possible that public-key quantum money intrinsically requires a new mathematical leap of faith, just as public-key cryptography required a new leap of faith when it was first introduced in the 1970s.”

That’s a surprising admission and a challenge.

But there’s another fly in the ointment for any scheme that depends for its security on the inability to carry out a calculation in polynomial time: it’s only secure under attack from conventional computers.

The trouble is that quantum mechanics may allow these kind of problem to be solved easily. Whatever mathematical leap of faith these authors are hoping for, it may just be the case that quantum money will only be “collision free” until quantum mechanics begins to play a significant role in processing information.

Ref:arxiv.org/abs/0912.3825: Breaking and Making Quantum Money: Toward a New Quantum Cryptographic Protocol

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