Harmonic sequence state-preparation

TL;DR

An efficient circuit to prepare a quantum state with amplitudes proportional to a harmonic sequence is demonstrated by first preparing a large quantum state with linearly related amplitudes and then applying a quantum Fourier transform.

Abstract

We demonstrate an efficient circuit to prepare a quantum state with amplitudes proportional to a harmonic sequence. We do this by first preparing a large quantum state with linearly related amplitudes and then applying a quantum Fourier transform; this has a direct analogy to the fact that the Fourier coefficients of a sawtooth wave follow a harmonic sequence. We then consider an extension of this problem by block-encoding a matrix with a harmonic sequence along its diagonal. The cost of both circuits is dominated by the costs associated with the quantum Fourier transform.

Figures (20)

• Figure 1: (Left) A state-preparation circuit $U$ for $|\psi\rangle$ using $a_c$ clean ancilla and $a_p$ persistent ancilla, resulting in an approximation state $|\tilde{\psi}\rangle$. (Right) A block-encoding circuit $V$ for $A$ using $a_c$ clean ancilla and $a_p$ persistent ancilla applied to a data register state $|\psi\rangle$, resulting in the state $\tilde{A}|\psi\rangle /\lVert\tilde{A}|\psi\rangle\rVert$.
• Figure 2: Decomposition of a controlled-Hadamard gate into Clifford + T using two T-gates.
• Figure 3: Plot of the amplitudes during the circuit in Figure \ref{['harmonicfig']} which transforms the cotangent state $|c\rangle$ to eliminate the constant real amplitude on the target state (i.e., the bottom $n$ qubits of an $n+m$-qubit state). The functions are depicted as continuous with discontinuity jumps where relevant. Red corresponds to imaginary part and blue corresponds to real part. The interval to the first dotted line corresponds to measuring the top $m$ qubits as $|0\rangle ^{\otimes m}$ (refer to this as the first interval). The interval from the start to the second dotted line corresponds to measuring the top qubit as $|0\rangle$. The interval from the third dotted line to the end corresponds to measuring the top $m$ qubits as $|1\rangle ^{\otimes m}$ (refer to this as the last interval). (a) The initial cotangent state $|c\rangle$ resulting from applying a quantum Fourier transform to the linear state $|L\rangle$. There is constant real part except at $|0\rangle ^{\otimes (n+m)}$ where both real and imaginary amplitudes vanish. Notice that positive imaginary amplitudes in the first interval cover the set $\{1,...,1/(N-1)\}$ whereas those in the last interval cover the set $\{ 1,...,1/N\}$ (i.e. there is an extra amplitude we need to swap out). We highlight the vanishing imaginary amplitude at the basis state $|1\rangle |0\rangle ^{\otimes (n+m-1)}$; this will eventually replace the $1/N$ amplitude. (b) We apply a series of multi-controlled $X$ gates on the previous state, open controls on the bottom $n$ qubits, closed control on the top qubit, $X$ gates everywhere else. This replaces the $1/N$ imaginary amplitude at $|1\rangle ^{\otimes m}|0\rangle ^{\otimes n}$ with the vanishing imaginary amplitude at $|1\rangle |0\rangle ^{\otimes (n+m-1)}$. (c) A series of CNOT gates with closed control on the top qubit flips the second half of the state. This changes the location of the last interval such that it can interact with the first interval simply by operating on the top qubit. (d) We apply an incrementer on the bottom $n$ qubits controlled on the top qubit. This ensures the imaginary amplitudes of the last interval directly match with those from the first interval but carry a negative sign. (e) $HX$ applied to the top qubit constructively interferes the asymptotes to the first interval and destructively interferes the asymptotes to the last interval. Most importantly, the real amplitudes destructively interfere in the first interval, which was the desired outcome.
• Figure 4: State-preparation circuit for a 5-qubit harmonic sequence state with 5 ancilla qubits.
• Figure 5: (Left) Expected T-depth of the harmonic sequence state-preparation as a function of number of data qubits $n$ and error $\epsilon$. (Center) Expected T-count of the harmonic sequence state-preparation scheme. A parallelism of $12\times$ can be observed without the use of phase gradient QFTs which is possible in most fault-tolerant architectures. With phase kickback, more favorable T-counts are possible. (Right) Total number of qubits (including all necessary ancilla for Toffolis, cotangent approximation, and RUS Clifford + T synthesis).

Theorems & Definitions (10)

• Proposition 1
• Proposition 2
• Proposition 3
• Lemma 1
• Lemma 2
• Lemma 3
• Proposition 4
• Lemma 4
• Lemma 5
• Lemma 6