HW3 - Problem 3
Problem 3: Quantum Money
Banker
Suppose you are a bank that is printing quantum money. Let’s begin by going through an example of what this may look like for serial codes with length \(n = 3\).
In our bank, we will encode our quantum states using the function \(f(x) = (x \cdot 17 + 4) (\text{mod }64)\), where \(x\) is the number that the binary serial code is representing.
- What does the serial code \(010\) get mapped to using this function?
- What does the serial code \(110\) get mapped to using this function?
- If a client handed you a bill with the serial number \(010\), what basis would you measure each qubit in to verify that it is in the correct state?
- If a client handed you a bill with the serial number \(110\), what basis would you measure each qubit in to verify that it is in the correct state?
Tricking the Bank
Suppose that after every verification process, the bank returns the bill to you (yes this is not smart, and we will formally show why in this problem), regardless of whether it passed the verification or not. They also allow you to submit your bill multiple times. Let’s design a scheme to counterfeit this money scheme to make a fake bill that passes the verification.
A quantum bill in this scheme can generally be written as a tensor product of \(n\) qubits,
\(\left|\psi\right> = \left|\psi_1\right> \otimes \left|\psi_2\right> \otimes \cdots \otimes \left|\psi_n\right>,\)
where each \(\left|\psi_i\right>\) is in one of the four states \(\{\left|0\right>, \left|1\right>, \left|+\right>, \left|-\right>\}\).
- Suppose we replace \(\left|\psi_1\right>\) with a qubit in the state \(\left|0\right>\). Assuming that the \(\left|\psi_1\right>\)$ could be any of the four possible states with equal probability, what is the probability that the verification was successful?
- Let \(\left|\psi\right>\) be the state received after the bank runs their verification process. If the verification was successful, what is the state of \(\left|\psi\right>\)?
- If the verification was unsuccessful, what is the state of \(\left|\psi\right>\)? You can use the shorthand \(\left|\psi_i^\perp\right>\) to represent a state orthogonal to \(\left|\psi_i\right>\).
- Design a counterfeiting strategy to create a copy of \(\left|\psi\right>\) for all \(n\) qubits. How many times would we need to resubmit to the bank?