Showing that Matrix Inversion is BQP-complete - HHL Algorithm

Question

I am trying to understand an argument that Matrix Inversion is BQP-complete for certain conditions on the matrix. This is explained here on page 39 (this paper is a primer to the HHL algorithm and gives some more detailed calculations, more detail about assumptions for people new to the subject).

Definition:

An algorithm solves matrix inversion if it has:

Input: An $O(1)$-sparse matrix hermitian $A$ of dimension $N$ specified using an oracle or via a $\mathrm{poly}(\log N)$-time algorithm that returns the non-zero elements of a row. The singular values of $A$ lie between $\frac{1} {\kappa} $ and 1, where $\kappa$ is the ration between the largest eigenvalue and the smallest eigenvalue of $A$.

Output:

A bit that equals one with probability $\langle x | M |x \rangle \pm \epsilon $ where $M = |0 \rangle \langle 0 | \otimes I^{N/2}$ corresponds to measuring the first qubit and $|x \rangle$ is a normalised state proportional to $A^{-1} |b \rangle$ for $|b \rangle = |0 \rangle$.

Let $C$ be a quantum circuit acting on $n = logN$ qubits which applies $T$ two-qubit gates $U_1, \cdots U_T$. The initial state is given by $|0 \rangle^{\otimes n}$, and the answer will be determined by measuring the first qubit of the final state. Adjoin an ancilla register of dimension $3T$ and define a unitary operation: $$U = \sum_{t-1}^{T}|t+1 \rangle \langle t| \otimes U_t + |t+T+1 \rangle \langle t+T| \otimes I $$$$ + |t+2T+1 \bmod 3T \rangle\langle t+2T| \otimes U^{\dagger}_{3T+1-t}.$$

The author then writes: "This operator has been chosen such that for $T+1 \leq t \leq 2T$, applying $U^t$ to the state $|1 \rangle |\psi \rangle$ yields the output state $|t+1 \rangle \otimes U_t \cdots U_1 |\psi \rangle$. We can see this as the first $T+1$ applications of $U$ return $|T+2 \rangle \otimes U_T \cdots U_1 |\psi \rangle$ . We see from the second term of the definition for $U$ that for the next $t'<T-1$ applications, the action on the $|\psi \rangle$ register remains unchanged, while the ancillary variable is merely being incremented."

This last statement is quite a mouthful as the expression is very complicated. I am having difficulty seeing why this is true and any insights would be much appreciated.

Answer 1

Define the states $$ |\psi_t\rangle=\left\{\begin{array}{cc} |t\rangle\otimes(U_{t-1}U_{t-2}\ldots U_1|\psi\rangle) & t=1,2,\ldots T \\ |t\rangle\otimes(U_{T}U_{T-1}\ldots U_1|\psi\rangle) & t=T+1,T+2,\ldots 2T \\ |t\rangle\otimes(U_{3T+1-t}U_{3T-t}\ldots U_1|\psi\rangle) & t=2T+1,2T+2,\ldots 3T \end{array}\right. $$ Now let $$ U=\frac{2}{T}\sum_{t=1}^{T}|t+1\rangle\langle t|\otimes U_t+|t+T+1\rangle\langle t+T|\otimes I+|t+2T+1\text{ mod }3T\rangle\langle t + 2T|\otimes U^\dagger_{3T+1-t}. $$ You can verify that $$ U|\psi_t\rangle=|\psi_{t+1}\rangle $$ for $t=1,2,\ldots 3T-1$ and $$ U|\psi_{3T}\rangle=|\psi_1\rangle. $$ So, basically, $U$ acts as a cyclic permutation through the states $\{|\psi_t\rangle\}$. Act $U$ $k$ times, (i.e. apply $U^k$) and you go from any $|\psi_t\rangle$ to $|\psi_{t+k}\rangle$.

Now, if you start in $|\psi_1\rangle$ and enact any $U^k$ for $k=T,T+1,\ldots 2T-1$, you get an outcome for which the second register is the same, $$ U_{T}U_{T-1}\ldots U_1|\psi\rangle. $$ This idea is hence that the $U_1$, $U_2$ etc can be thought of as the individual unitaries of an arbitrary quantum computation of $T$ steps, and that this state is the overall output of the computation.

The paper then goes on to talk about implementing $$ A=I-Ue^{-1/T}, $$ and it's $A^{-1}$ you're trying to calculate. Using a Taylor expansion, this would be $$ A^{-1}=\sum_tU^te^{-t/T} $$ So, this includes $t=T$ to $2T-1$ with high (enough) probability, meaning that the act of the inversion effectively gives us the outcome of a quantum computation. If it can calculate the act of any quantum computer, the computation is at least as hard as the hardest thing a quantum computer can calculate, so the computation is BQP-hard. Since it can be completed on a quantum computer, the calculation is in BQP, and hence the overall classification is BQP-complete.