Probabilistic state synthesis based on optimal convex approximation

TL;DR

It is demonstrated that the optimal probabilistic synthesis quadratically reduces the approximation error and it is shown that a deterministic synthesis algorithm can be efficiently converted into a probabilistic one that achieves this quadratic error reduction.

Abstract

When preparing a pure state with a quantum circuit, there is an unavoidable approximation error due to the compilation error in fault-tolerant implementation. A recently proposed approach called probabilistic state synthesis, where the circuit is probabilistically sampled, is able to reduce the approximation error compared to conventional deterministic synthesis. In this paper, we demonstrate that the optimal probabilistic synthesis quadratically reduces the approximation error. Moreover, we show that a deterministic synthesis algorithm can be efficiently converted into a probabilistic one that achieves this quadratic error reduction. We also numerically demonstrate how this conversion reduces the $T$-count and analytically prove that this conversion halves an information-theoretic lower bound on the circuit size. In order to derive these results, we prove general theorems about the optimal convex approximation of a quantum state. Furthermore, we demonstrate that this theorem can be used to analyze an entanglement measure.

Figures (5)

• Figure 1: Quadratic reduction of the approximation error by using probabilistic synthesis. We assume that we can exactly generate an eigenstate $\hat{\phi}_x$ of the Pauli operators, represented by the six extreme points of the octahedron. We represent the Bloch sphere by a sphere with radius $\frac{1}{2}$, where the trace distance between two quantum states equals the Euclidean distance between the corresponding points. (a) We can compute $\min_p\left\|\phi-\sum_xp(x)\hat{\phi}_x\right\|_{\text{tr}}=\epsilon^2=\frac{1}{2\sqrt{3}}\left(\sqrt{3}-1\right)$ and $\min_x\left\|\phi-\hat{\phi}_x\right\|_{\text{tr}}=\epsilon$, where $\phi$ is the furthest state from $\{\hat{\phi}_x\}_{x=1}^6$, represented as a large red point. (b) Suppose that the target state is chosen from $S_G:=\{\phi:| {\phi} \rangle=\cos t| {0} \rangle+\sin t| {1} \rangle,t\in\mathbb{R}\}$, represented by a meridian. We can compute $\min_p\left\|\phi-\sum_xp(x)\hat{\phi}_x\right\|_{\text{tr}}=\tilde{\epsilon}^2=\frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\right)$ and $\min_{x}\left\|\phi-\hat{\phi}_x\right\|_{\text{tr}}=\tilde{\epsilon}$, where $\phi$ is the furthest state in $S_G$ from $\{\hat{\phi}_x\}_{x=1}^6$, represented as a large red point.
• Figure 2: Probabilistic encoding of pure state $\phi$ on a $d$-dimensional system using $n$-bit strings and a decoder $\Gamma$ that generates an approximated state $\hat{\rho}$. State $\phi$ is probabilistically encoded in label $x$ in a finite set $X$ in accordance with probability distribution $p_\phi:X\rightarrow[0,1]$. As a special case of probabilistic encoding, we also consider deterministic encoding that utilizes probability distribution $p_\phi:X\rightarrow\{0,1\}$. Note that the length of classical bit strings to represent $x\in X$ is given by $n=\lceil\log_2|X|\rceil$.
• Figure 3: Relationship between $T$-count and the approximation error for synthesizing $| {\phi} \rangle=\cos t| {0} \rangle+\sin t| {1} \rangle$ with $t=1$. For each target approximation error, we run the Ross-Selinger algorithm to obtain a gate sequence to approximate $\phi$. The blue dashed line interpolates points, each of which represents a target approximation error and the $T$-count of the gate sequence. The actual approximation error and the $T$-count achieved by the gate sequence are plotted by blue dots. Note that both the target and actual approximation errors are represented by $\epsilon$. For each of the target approximation errors, we run the probabilistic synthesis algorithm and obtain a list of six gate sequences to be probabilistically sampled. The purple dashed line interpolates points, each of which represents a target approximation error and the maximum $T$-count of gate sequences in the list. The actual approximation error and the maximum $T$-count achieved by optimally mixing the gate sequence are plotted by purple dots.
• Figure 4: Relationship between $| {\psi} \rangle$, $| {\phi} \rangle$, $| {\hat{\phi}} \rangle$ and $| {\hat{\phi}_x} \rangle$.$| {\psi} \rangle$, $| {\phi} \rangle$, $| {\hat{\phi}} \rangle$ and $| {\hat{\phi}_x} \rangle$ correspond to $\cos(t_1-t_2)\sin(t_1-t_2)$, $\cos t_2-\sin t_2$, $10$ and $a\text{Re}\left(\beta\right)$, respectively.
• Figure 5: Plots of estimated values of $\mu(B_{\frac{1}{2}}\left(\rho\right))$ (dots) and $g_{4,\frac{1}{2}}(p_0)$ (curve) for $\rho=p_0| {0} \rangle\langle {0} |+(1-p_0)| {1} \rangle\langle {1} |\in\mathbf{S}\left(\mathbb{C}^{4}\right)$.$\mu(B_{\frac{1}{2}}\left(\rho\right))$ is estimated by uniformly sampling $10^7$ pure states. The plots indicate $\mu(B_{\frac{1}{2}}\left(\rho\right))$ is accurately upper bounded by $g_{4,\frac{1}{2}}(p_0)$.

Theorems & Definitions (23)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• proof : Proof of Lemma \ref{['lemma:worstcase']}
• Theorem 1
• Proposition 1
• proof
• Lemma 4