1.1 How many basis states are there for a three-qubit system? Show the vectors for the (computational) basis states.

       1.2 There are four Bell states that can be created by entangling two qubits. In addition to the state shown in Figure 1.5, the three additional states are listed below. Construct a circuit for generating each state.12(|00⟩−|11⟩)12(|01⟩+|10⟩)12(|01⟩−|10⟩)

       1.3 Prove the following equivalencies.HZH = XHXH = ZHYH=−YCNOT1,0=H⊗2CNOT0,1H⊗2 In (d), CNOTi,j means a CNOT with qubit i as the control and qubit j as the target. H⊗2 means a Hadamard gate applied to both qubits.

       1.4 Create a quantum circuit that swaps two qubit states. In other words: |ab⟩ ↦ |ba⟩. Hint: Consider this classical algorithm that swaps two numbers x and y using an exclusive-OR (XOR) instruction.

       1.5 Suppose we design a superconducting qubit where the energy difference between |0⟩ and |1⟩ is around 10 GHz. What is the temperature needed to minimize the effect of thermal energy on the qubit, assuming that the qubit is in thermal equilibrium with its environment?

       1.6 Consider the following three-qubit quantum state:Is qubit q2 entangled with the other two qubits? Explain why or why not.

       1.7 Suppose we have a way of measuring a qubit in the |+⟩ and |−⟩ basis. As a reminder:Given a qubit in the |0⟩ state, what is the probability of measuring |+⟩? (Hint: Use the Born Rule.)Given a qubit in the |1⟩ state, what is the probability of measuring |+⟩?Given a qubit |R⟩=12|0⟩+i2|1⟩, what is the probability of measuring |−⟩?

       1.8 Given a qubitWhat is the probability of measuring |0⟩?What is the probability of measuring |+⟩?What is the probability of measuring |−⟩?

      1 1 To be more precise, |0⟩ and |1⟩ are known as the “computational basis” or “standard basis” states. For this chapter, we will restrict our discussion to the standard basis and will simply refer to basis states.

      2 2 Note: There are classical multi-input logic gates that are reversible, called “conservative logic” gates. To date these have not been widely used, but they have the advantage of making it possible in principle to compute without dissipating power [1].

      3 3 Entanglement can be used to increase the capacity of a communication channel, however, using a protocol known as super-dense coding.

      4 4 The transpose of a matrix or vector is formed by interchanging the rows and columns. The transpose of a column vector gives a row vector. The transpose complex conjugate is formed by taking the transpose of the vector or matrix and then taking the complex conjugate of each element.