Quadratic improvement on accuracy of approximating pure quantum states and unitary gates by probabilistic implementation

Pure quantum states are often approximately encoded in classical states in various quantum information processing tasks, such as classical bit strings storing probability amplitudes of pure states in the classical simulation of quantum circuits; classical data obtained by measurement in state tomography or state estimation; classical descriptions of quantum circuits that generate target pure states in quantum circuit synthesis [1, 2]. Recently, compact classical encodings from which one can predict probability distributions on outcomes obtained by measurements in certain classes have been developed [3, 4, 5].

The key issue is the minimum encoding. Here, we investigate it in terms of the minimum length $n$ of classical bit strings, with which one can approximately achieve any information processing task achievable with the original pure state on a $d$-dimensional system. That is, from the classical strings, one can construct a quantum state $\hat{\rho}$ that is indistinguishable from the original state $\phi$ within a certain accuracy by any measurements. In addition to deterministic encodings, which deterministically associate $\phi$ with a classical bit string, we consider probabilistic encodings, which associate $\phi$ with one of multiple classical strings according to some distribution. That is, in probabilistic encodings, $\phi$ is associated with an ensemble of classical strings from which $\hat{\rho}$ is constructed as an ensemble of the quantum states encoded in the classical strings.

Besides providing fundamental limits for information processing tasks using classical encodings, the minimum length $n$ is a fundamental quantity in various theoretical subjects including communication complexity, computational complexity, and asymptotic geometric analysis. Indeed, in deterministic encodings, classical strings do nothing but encode elements in an $\epsilon$-covering (sometimes called an $\epsilon$-net) of the set of pure states. Thus, the minimum bit length $n$ is the logarithm of the minimum size of $\epsilon$-coverings (called the covering number). Due to its prominent role in algorithm design and asymptotic geometric analysis, the covering number has been well-studied [6, 7, 8], and it is known that $n=O(d)$ bits are enough to encode an $\epsilon$-covering. On the other hand, a particular task in communication complexity called distributed quantum sampling, which aims to classically transmit a pure state so as to approximately sample outcomes of arbitrary quantum measurement, provides a lower bound on the minimum length required for probabilistic encodings as $n=\Omega(d)$ [9]. Taking the two known facts into account, it seems that the minimum probabilistic encoding can be realized by a deterministic one. This intuition is supported by the fact that an ensemble of deterministic classical states (called a probabilistic classical state) represents our lack of knowledge about a classical system while a pure state represents our maximum knowledge about a quantum system.

Contrary to this intuition, we show that the minimum length $n$ required for probabilistic encodings is exactly half of the one required for deterministic encodings in the asymptotic limits of the dimension or accuracy. Thus, the minimum encoding must associate some pure states with ensembles of classical strings describing distinct quantum states, which may be counter-intuitive, considering that pure states themselves are not probabilistic mixtures of distinct quantum states. The excessive bits required for deterministic encodings can be interpreted as a consequence of their excessive predictive capability such that they can not only reconstruct quantum states within a certain accuracy but also deterministically compute the probability distribution of any measurements within the same accuracy. Such a deterministic computation is impossible by using either the minimum (probabilistic) encoding or the original quantum states.

The bit length reduction by using probabilistic encodings follows from our refined estimation of the covering number and the following fact we prove: for any finite set $\mathcal{A}$ of pure states, it is possible to quadratically increase the accuracy of approximating arbitrary pure states by using ensembles of pure states in $\mathcal{A}$. Moreover, we show that a similar fact holds when we consider the approximation of unitary gates by using a finite set of unitary gates. Recently, it has been found that when we approximately implement arbitrary unitary gates by using a gate sequence over a finite universal gate set (called circuit synthesis), the length of the gate sequence or the number of $T$ gates can be reduced by using ensembles of gate sequences for high-accuracy circuit synthesis [10, 11]. Our result improves the reduction rate of the previous results and shows that the reduction is possible even for low-accuracy circuit synthesis, which might improve the accuracy of NISQ algorithms.

In this section, we summarize basic notations used throughout the paper. Note that we consider only finite-dimensional Hilbert spaces. In particular, two-dimensional Hilbert space $\mathbb{C}^{2}$ is called a qubit. $\mathbf{L}\left(\mathcal{H}\right)$ and $\mathbf{Pos}\left(\mathcal{H}\right)$ represent the set of linear operators and positive semidefinite operators on Hilbert space $\mathcal{H}$, respectively. $\mathbf{S}\left(\mathcal{H}\right):=\left\{\rho\in\mathbf{Pos}\left(\mathcal{H}\right):\text{tr}\left[\rho\right]=1\right\}$ and $\mathbf{P}\left(\mathcal{H}\right):=\left\{\rho\in\mathbf{S}\left(\mathcal{H}\right):\text{tr}\left[\rho^{2}\right]=1\right\}$ represent the set of quantum states and that of pure states, respectively. Pure state $\phi\in\mathbf{P}\left(\mathcal{H}\right)$ is sometimes alternatively represented by complex unit vector $|{\phi}\rangle\in\mathcal{H}$ satisfying $\phi=|{\phi}\rangle\langle{\phi}|$. Any physical transformation of the quantum state can be represented by a completely positive and trace preserving (CPTP) linear mapping $\Gamma:\mathbf{L}\left(\mathcal{H}\right)\rightarrow\mathbf{L}\left(\mathcal{H}^{\prime}\right)$.

The trace distance $\left\|\rho-\sigma\right\|_{\text{tr}}$ of two quantum states $\rho,\sigma\in\mathbf{S}\left(\mathcal{H}\right)$ is defined as $\left\|M\right\|_{\text{tr}}:=\frac{1}{2}\text{tr}\left[\sqrt{MM^{\dagger}}\right]$ for $M\in\mathbf{L}\left(\mathcal{H}\right)$. It represents the maximum total variation distance between probability distributions obtained by measurements performed on two quantum states. A similar notion measuring the distinguishability of $\rho$ and $\sigma$ is the fidelity function, defined by $F\left(\rho,\sigma\right):=\max\text{tr}\left[\Phi^{\rho}\Phi^{\sigma}\right]$, where $\Phi^{\rho}\in\mathbf{P}\left(\mathcal{H}\otimes\mathcal{H}^{\prime}\right)$ is a purification of $\rho$, i.e., $\rho=\text{tr}_{\mathcal{H}^{\prime}}\left[\Phi^{\rho}\right]$, and the maximization is taken over all the purifications. Fuchs-van de Graaf inequalities [12] provide relationships between the two measures with respect to the distinguishability as follows:

holds for any states $\rho,\sigma\in\mathbf{S}\left(\mathcal{H}\right)$, where the equality of the right inequality holds when $\rho$ and $\sigma$ are pure.

The existence of a probabilistic encoding is equivalent to the existence of physical transformation $\Gamma$ that can approximately reconstruct arbitrary pure state $\phi$ from probabilistic classical state $\{(p_{\phi}(x),x)\}_{x\in X}$ as shown in Fig. 1. Physical transformation $\Gamma$ can be regarded as a decoder of the classical states to quantum states, which outputs mixed state $\rho_{x}\in\mathbf{S}\left(\mathbb{C}^{d}\right)$ when label $x\in X$ is inputted. Formally, $\Gamma$ is represented by a classical-quantum channel [13], which is defined as $\Gamma(\sigma)=\sum_{x\in X}\langle{x}|\sigma|{x}\rangle\rho_{x}$, where $\left\{|{x}\rangle\in\mathbb{C}^{|X|}\right\}_{x\in X}$ is an orthonormal basis. We require that with the output state $\hat{\rho}:=\sum_{x}p_{\phi}(x)\rho_{x}$ of $\Gamma$, one can approximately sample any measurement outcomes performed on $\phi$ within total variation distance $\epsilon$, i.e., $\left\|\phi-\hat{\rho}\right\|_{\text{tr}}<\epsilon$ for given $\epsilon\in(0,1]$. Thus, we say that a probabilistic encoding of $\mathbf{P}\left(\mathbb{C}^{d}\right)$ with accuracy $\epsilon$ exists if and only if there exists set $\{\rho_{x}\in\mathbf{S}\left(\mathbb{C}^{d}\right)\}_{x\in X}$ of quantum states satisfying

where the minimum is taken over probability distribution $p$ over $X$, i.e, $\sum_{x\in X}p(x)=1$ and $p(x)\geq 0$. Note that since any mixed state is a probabilistic mixture of pure states and trace distance is convex, Eq. (2) also guarantees that an arbitrary mixed state is also approximately reconstructible within the same accuracy as pure states.

In the context of circuit synthesis using a finite universal gate set, it has recently been found that the length of the gate sequence or the number of $T$ gates can be reduced by using ensembles of gate sequences [10, 11]. This is based on the so-called mixing lemma, which shows that if finite set $\{\Upsilon_{x}\}_{x}$ of unitary transformations approximates arbitrary unitary transformations within sufficient high accuracy, it is possible to increase the accuracy by using particular ensembles $\sum_{x}p(x)\Upsilon_{x}$.

Based on Corollary 1, where we derive the accuracy achieved by the optimal ensembles of pure states in approximating arbitrary pure states, we can derive bounds on the accuracy achieved by the optimal ensembles of unitary transformations in approximating arbitrary unitary transformations in the following theorem. Note that similarly to [10, 11], we measure the accuracy of the approximation by using the diamond norm (sometimes called the completely bounded trace norm) of hermitian preserving linear mappings, defined as

where $\Xi:\mathbf{L}\left(\mathcal{H}_{1}\right)\rightarrow\mathbf{L}\left(\mathcal{H}_{2}\right)$, $id_{\mathcal{H}}$ is the identity mapping on $\mathbf{L}\left(\mathcal{H}\right)$ and $\dim\mathcal{H}_{1}^{\prime}=\dim\mathcal{H}_{1}$.

This theorem resembles Corollary 1 (especially when $d=2$), which can be regarded as a consequence of the similarity between pure states and unitary transformations via the Choi-Jamiołkowski representation. Although a proof uses the minimax theorem as in the proof of Corollary 1, it requires additional work to some extent. We give a complete proof in Appendix D with the sharp lower bound.

This theorem implies that quantum circuit synthesis using probabilistically generated circuits formed from finite universal gate set $\mathcal{C}=\{g_{1},g_{2},\cdots\}$ can reduce the circuit size. To see this, first, let $\{\Upsilon_{x}^{(n)}\}_{x}$ be the set of unitary transformations representing the unitary circuit realized by gate sequences $g_{i_{1}}\cdots g_{i_{n}}$ of length at most $n$. Next, let $n(\epsilon)$ be the smallest length of gate sequences to approximate arbitrary unitary transformations within accuracy $\epsilon$, i.e., $n(\epsilon):=\min\{n\in\mathbb{N}:\max_{\Upsilon}\min_{x}\frac{1}{2}\left\|\Upsilon-\Upsilon_{x}^{(n)}\right\|_{\diamond}<\epsilon\}$. Theorem 3 implies that by using the probabilistic implementation, we can implement arbitrary unitary transformation within accuracy $\epsilon$ only with an $n(\sqrt{\epsilon})$-size circuit.

The accuracy in approximating unitary transformations is often measured by using the operator norm $\left\|X\right\|_{\infty}:=\max_{\phi\in\mathbf{P}\left(\mathcal{H}\right)}\left\|X|{\phi}\rangle\right\|_{2}$. The celebrated Solovay-Kitaev theorem shows that for any finite universal gate set $\mathcal{C}$, set $\{U_{x}^{(n)}\}_{x}$ of unitary circuits realized by gate sequences of length $n=O\left(\log^{c}\left(\frac{1}{\delta}\right)\right)$ ($c\geq 1$) is sufficient for approximating arbitrary unitary operators, i.e., $\max_{U}\min_{x}\left\|U-U_{x}^{(n)}\right\|_{\infty}<\delta$, where we denote a unitary circuit and the unitary operator representing the circuit by the same symbol $U_{x}^{(n)}$. On the other hand, a relationship between the operator norm and the diamond norm shown in Appendix E implies that $\left\|\Upsilon-\Upsilon_{x}\right\|_{\diamond}<\delta\sqrt{4-\delta^{2}}$ if $\left\|U-U_{x}\right\|_{\infty}<\delta$, where $\Upsilon(\rho)=U\rho U^{\dagger}$ and $\Upsilon_{x}(\rho)=U_{x}\rho U_{x}^{\dagger}$. Combining with Theorem 3, we can verify that arbitrary unitary transformations can be approximated by ensembles of $\{\Upsilon_{x}\}_{x}$ such that

if $\max_{U}\min_{x}\left\|U-U_{x}\right\|_{\infty}<\delta$. This bound is tighter than the previous bound obtained in [10, Theorem 1], $\max_{\Upsilon}\min_{p}\frac{1}{2}\left\|\Upsilon-\sum_{x}p(x)\Upsilon_{x}\right\|_{\diamond}<5\delta^{2}$. Moreover, our bound holds if $\delta\leq\sqrt{2}$ while the previous bound was shown to hold for $\delta<0.01$. In addition to the improved estimation, our lower bound reveals the limitation of the probabilistic implemetation.

As suggested by the Solovay-Kitaev theorem, if we assume $n(\epsilon)\sim a\log^{c}\left(\frac{1}{\epsilon}\right)$ in the high-accuracy regime ($\epsilon\ll 1$), the probabilistic implementation reduces the length of gate sequences by about $1-\left(\frac{1}{2}\right)^{c}\geq 50\%$ since $c\geq 1$ from a volume consideration. Even in the low-accuracy regime, our bound guarantees the reduction. For example, we show how the gate length can be reduced by using the probabilistic implementation to synthesize single-qubit unitary transformations with gate set $\{S,H,T\}$ in Appendix F.

In this paper, we have considered the minimum probabilistic encoding so as to approximately reconstruct an arbitrary pure state $\phi\in\mathbf{P}\left(\mathbb{C}^{d}\right)$ from an $n$-bit string within accuracy $\epsilon$ with respect to the trace distance. We then demonstrated that it cannot be realized by simply storing an element of the minimum $\epsilon$-covering. More precisely, we proved that the bit rate required for probabilistic encodings is exactly half of that of the minimum length of bits necessary to store elements of an $\epsilon$-covering of $\mathbf{P}\left(\mathbb{C}^{d}\right)$ in asymptotic limit $\epsilon\rightarrow 0$ or in limit $d\rightarrow\infty$ when $\epsilon\in\left(0,\frac{1}{2}\right]$. In limit $d\rightarrow\infty$ when $\epsilon\in\left(\frac{1}{2},1\right]$, the same result holds if we consider only internal $\epsilon$-coverings; however, in general, whether the same result holds or not is an open problem. Several numerical calculations suggest a positive answer.

Moreover, we show that similarly to the state encoding, for any finite set $\{\Upsilon_{x}\}_{x}$ of unitary transformations, we can at least quadratically increase the accuracy of approximating arbitrary unitary transformations by using ensembles of $\{\Upsilon_{x}\}_{x}$. Particularly, we obtain bounds on the accuracy when one uses the optimal ensembles to reveal the possibility and the limitation of the probabilistic implementation of unitary transformations.

Our result could provide a new quantitative guiding principle to explore further capabilities and limitations of manipulating a quantum system as well as the foundations of quantum theory, including the following two related topics:

1. 1.

   The results demonstrate an information-theoretical separation of the memory size to store a pure quantum state between strong simulations and weak ones, which are two types of classical simulation of a quantum computer [14, 15, 16] (the former approximately computes the probability distribution over the outcomes, whereas the latter only approximately samples the outcomes.)

2. 2.

   The complex projective space representation of pure states can be regarded as a classical encoding of pure states in deterministic classical states describing operators in $\mathbf{P}\left(\mathcal{H}\right)$. The fact that any distinct pure states are encoded indistinguishable classical states inclines us to think that the indistinguishability of non-orthogonal pure states results from our limited ability to measure them. To interpret the indistinguishability as an intrinsic feature of pure states, classical encodings of pure states in probabilistic classical states have been constructed in $\psi$-epistemic models [17, 18], in which the encodings use indistinguishable and probabilistic classical states to encode some distinct pure states. Our results show that indistinguishable classical states encoding distinct elements in an $\epsilon$-covering are not only helpful for such an interpretation but also necessary for the minimum probabilistic encoding.