Efficient classical simulation of Clifford circuits with nonstabilizer input states

TL;DR

This paper presents efficient classical algorithms for approximating output probabilities of Clifford circuits with nonstabilizer inputs, especially when inputs are mixed or pure nonstabilizer states, under certain restrictions.

Contribution

It introduces new algorithms that efficiently approximate output probabilities for Clifford circuits with nonstabilizer inputs, expanding classical simulation capabilities.

Findings

Efficient approximation algorithms for mixed input states.

Approximation algorithms for pure nonstabilizer product states with restrictions.

Applications to Clifford circuits with magic states, PBC, and IQP circuits.

Abstract

We investigate the problem of evaluating the output probabilities of Clifford circuits with nonstabilizer product input states. First, we consider the case when the input state is mixed, and give an efficient classical algorithm to approximate the output probabilities, with respect to the $l_1$ norm, of a large fraction of Clifford circuits. The running time of our algorithm decreases as the inputs become more mixed. Second, we consider the case when the input state is a pure nonstabilizer product state, and show that a similar efficient algorithm exists to approximate the output probabilities, when a suitable restriction is placed on the number of qubits measured. This restriction depends on a magic monotone that we call the Pauli rank. We apply our results to give an efficient output probability approximation algorithm for some restricted quantum computation models, such as Clifford circuits with solely magic state inputs (CM), Pauli-based computation (PBC) and instantaneous quantum polynomial time (IQP) circuits.

Equations

$$F(\psi )=\underset{|\varphi ⟩}{\max }|⟨\varphi |\psi ⟩{|}^2,$$

$$\mu (\psi ):=2(1-F(\psi )).$$

$$\rho =\frac{1}{2}\left({\rho }_{00}I+{\rho }_{10}X+{\rho }_{01}Z+{\rho }_{11}XZ\right),$$

$$X^b{Z}^a\rho Z^a{X}^b=\frac{1}{2}\sum _{s,t\in {\mathbb{F}}_2}(-1{)}^{sa+tb}{\rho }_{st}{X}^s{Z}^t.$$

$${\mathbb{E}}_{a\in {\mathbb{F}}_2,b\in {\mathbb{F}}_2}{X}^b{Z}^a\rho Z^a{X}^b(-1{)}^{sa+tb}=\frac{1}{2}{\rho }_{st}{X}^s{Z}^t.$$

$$q(\vec{y})=⟨\vec{y}|U|0⟩⟨0{|}^{\otimes n}{\otimes }_{i=1}^m{\rho }_i{U}^{†}|\vec{y}⟩,$$

$$q_{\vec{a},\vec{b}}(\vec{y})=⟨\vec{y}|U|0⟩⟨0{|}^{\otimes n}\otimes (X^{\vec{b}}{Z}^{\vec{a}}{\otimes }_{i=1}^m{\rho }_i{Z}^{\vec{a}}{X}^{\vec{b}})U^{†}|\vec{y}⟩=⟨\vec{y}|U|0⟩⟨0{|}^{\otimes n}\otimes ({\otimes }_{i=1}^m{X}^{b_i}{Z}^{a_i}{\rho }_i{Z}^{a_i}{X}^{b_i})U^{†}|\vec{y}⟩.$$

$${\hat{q}}_{\vec{s},\vec{t}}$$

$${\hat{q}}_{\vec{s},\vec{t}}=⟨\vec{y}|U|0⟩⟨0{|}^{\otimes n}{\otimes }_{i=1}^m{X}^{s_i}{Z}^{t_i}{U}^{†}|\vec{y}⟩\cdot \prod _{i=1}^m\left(\frac{{\rho }_{s_i{t}_i}^{(i)}}{2}\right),$$

$$U|0⟩⟨0{|}^{\otimes n}{\otimes }_{i=1}^m{X}^{s_i}{Z}^{t_i}{U}^{†}=\prod _{i=1}^n\left(\frac{I+P_i}{2}\right)\prod _{j=1}^m{Q}_j,$$

$$\sigma =\frac{1}{2}\left({\sigma }_{00}I+{\sigma }_{10}X+{\sigma }_{01}Z+{\sigma }_{11}XZ\right),$$

$${\rho }_{st}=(1-\lambda {)}^{w(s,t)}{\sigma }_{st},$$

$$w(s,t)=\{\begin{array}{ll}0,& s=0,t=0\\ 1,& \text{otherwise}\end{array}.$$

$${\rho }_{st}=\mathrm{Tr}\left[X^s\rho Z^t\right]=(1-\lambda )\mathrm{Tr}\left[X^s\sigma Z^t\right]+\lambda /2\mathrm{Tr}\left[X^s{Z}^t\right]=(1-\lambda ){\sigma }_{st}+\lambda {\delta }_{s,0}{\delta }_{t,0}=(1-\lambda {)}^{w(s,t)}{\sigma }_{st},$$

$$p(\vec{y})=⟨\vec{y}|U|0⟩⟨0{|}^{\otimes n}{\otimes }_{i=1}^m{\sigma }_i{U}^{†}|\vec{y}⟩.$$

$$p_{\vec{a},\vec{b}}(\vec{y})=⟨\vec{y}|U|0⟩⟨0{|}^{\otimes n}\otimes ({\otimes }_{i=1}^m{X}^{b_i}{Z}^{a_i}{\sigma }_i{Z}^{a_i}{X}^{b_i})U^{†}|\vec{y}⟩.$$

$${\hat{p}}_{\vec{s},\vec{t}}=⟨\vec{y}|U|0⟩⟨0{|}^{\otimes n}{\otimes }_{i=1}^m{X}^{s_i}{Z}^{t_i}{U}^{†}|\vec{y}⟩\cdot \prod _{i=1}^m\left(\frac{{\sigma }_{s_i{t}_i}^{(i)}}{2}\right),$$

$$|{\hat{q}}_{\vec{s},\vec{t}}|\le (1-\lambda {)}^{w(\vec{s},\vec{t})}|{\hat{p}}_{\vec{s},\vec{t}}|,$$

$$w(\vec{s},\vec{t}):=\sum _{i}w(s_i,t_i).$$

$${\mathbb{E}}_{U\sim \mathcal{C}{l}_n}{U}^{†}AUW{U}^{†}BU={\int }_{U(2^n)}\mathrm{d}U{U}^{†}AUW{U}^{†}BU,$$

$${\int }_{U(2^n)}\mathrm{d}U{U}^{†}AUW{U}^{†}BU=\frac{\mathrm{Tr}\left[AB\right]\mathrm{Tr}\left[W\right]}{2^n}\frac{I}{2^n}+\frac{2^n\mathrm{Tr}\left[A\right]\mathrm{Tr}\left[B\right]-\mathrm{Tr}\left[AB\right]}{2^n(2^{2n}-1)}\left(W-\mathrm{Tr}\left[W\right]\frac{I}{2^n}\right).$$

$${\hat{q}}_{\vec{s},\vec{t}}^{\prime }(\vec{y})=\{\begin{array}{ll}{\hat{q}}_{\vec{s},\vec{t}}(\vec{y}),& w(\vec{s},\vec{t})\le l\\ 0,& \text{otherwise}\end{array},$$

$${\Vert q_{\vec{0},\vec{0}}^{\prime }-q_{\vec{0},\vec{0}}\Vert }_1=\sum _{\vec{y}\in {\mathbb{F}}_2^{n+m}}|q_{\vec{0},\vec{0}}^{\prime }(\vec{y})-q_{\vec{0},\vec{0}}(\vec{y})|$$

$${\hat{q}}_{\vec{s},\vec{t}}(\vec{y})[U](-1{)}^{\vec{a}\cdot \vec{s}+\vec{b}\cdot \vec{t}}={\hat{q}}_{\vec{s},\vec{t}}(\vec{y})[U^{\prime }],$$

$${\mathbb{E}}_{U\sim \mathcal{C}{l}_{n+m}}{\Vert q_{\vec{0},\vec{0}}^{\prime }-q_{\vec{0},\vec{0}}\Vert }_1^2={\mathbb{E}}_{U\sim \mathcal{C}{l}_{n+m}}{\Vert q_{\vec{a},\vec{b}}^{\prime }-q_{\vec{a},\vec{b}}\Vert }_1^2={\mathbb{E}}_{U\sim \mathcal{C}{l}_{n+m}}{\mathbb{E}}_{\vec{a}\in {\mathbb{F}}_2^m,\vec{b}\in {\mathbb{F}}_2^m}{\Vert q_{\vec{a},\vec{b}}^{\prime }-q_{\vec{a},\vec{b}}\Vert }_1^2.$$

$${\mathbb{E}}_{\vec{a}\in {\mathbb{F}}_2^m,\vec{b}\in {\mathbb{F}}_2^m}{\Vert q_{\vec{a},\vec{b}}^{\prime }-q_{\vec{a},\vec{b}}\Vert }_1^2$$

$${\mathbb{E}}_{U\sim \mathcal{C}{l}_{n+m}}{p}_{\vec{a},\vec{b}}^2(\vec{y})\le 2\cdot 2^{-2(n+m)}.$$

$${\mathbb{E}}_{U\sim \mathcal{C}{l}_{n+m}}{\Vert q_{\vec{0},\vec{0}}^{\prime }-q_{\vec{0},\vec{0}}\Vert }_1^2\le 2e^{-\lambda l}.$$

$${\mathrm{Pr}}_{U\sim \mathcal{C}{l}_{n+m}}\left[{\Vert q_{\vec{0},\vec{0}}^{\prime }-q_{\vec{0},\vec{0}}\Vert }_1\le \sqrt{\alpha }{e}^{-\lambda l}\right]\ge 1-\frac{2}{\alpha }.$$

$$q_{\vec{a},\vec{b}}(\vec{y})=⟨\vec{y}|UV|0⟩⟨0{|}^{\otimes n}\otimes (X^{\vec{b}}{Z}^{\vec{a}}{\otimes }_{i=1}^m{\rho }_i{Z}^{\vec{a}}{X}^{\vec{b}})V^{†}{U}^{†}|\vec{y}⟩=⟨\vec{y}|UV|0⟩⟨0{|}^{\otimes n}\otimes ({\otimes }_{i=1}^m{X}^{b_i}{Z}^{a_i}{\rho }_i{Z}^{a_i}{X}^{b_i})U^{†}{V}^{†}|\vec{y}⟩.$$

Full text

I Introduction

One of the main motivations behind the field of quantum computation is the expectation that quantum computers can solve certain problems much faster than classical computers. This expectation has been driven by the discovery of quantum algorithms which can solve certain problems believed to be intractable on a classical computer. A famous example of such a quantum algorithm is due to Shor, whose eponymous algorithm can solve the factoring problem exponentially faster than the best classical algorithms we know today Shor (1994, 1999).

With the advent of noisy intermediate-scale quantum (NISQ) devices Preskill (2018), an important near-term milestone in the field is to demonstrate that quantum computers are capable of performing computational tasks that classical computers cannot, a goal known as quantum supremacy Preskill ; Harrow and Montanaro (2017). Several restricted models of quantum computation have been proposed as candidates for demonstrating quantum supremacy. These include boson sampling Aaronson and Arkhipov (2011), the one clean qubit model (DQC1) Knill and Laflamme (1998); Fujii et al. (2018), instantaneous quantum polynomial-time (IQP) circuits Bremner et al. (2010), Hadamard-classical circuits with one qubit (HC1Q) Morimae et al. (2018), Clifford circuits with magic initial states and nonadaptive measurements Jozsa and Van den Nest (2014); Koh (2017); Yoganathan et al. , the random circuit sampling model Boixo et al. (2018); Bouland et al. (2018a), and conjugated Clifford circuits (CCC) Bouland et al. (2018b). These models are potentially good candidates for quantum supremacy because they can solve sampling problems that are conjectured to be intractable for classical computers, and are conceivably easier to implement in experimental settings.

In contrast to the above models, quantum circuits with Clifford gates and stabilizer input states are not a candidate for quantum supremacy, because they can be efficiently simulated on a classical computer using the Gottesman-Knill simulation algorithm Gottesman (1999). The Gottesman-Knill algorithm, however, breaks down and efficient classical simulability can be proved to be impossible (under plausible assumptions) when Clifford circuits are modified in various ways, under various notions of simulation Jozsa and Van den Nest (2014); Koh (2017); Bouland et al. (2018b); Yoganathan et al. . For example, it can be proved under plausible complexity assumptions that no efficient classical sampling algorithm exists that can sample from the output distributions of Clifford circuits with general product state inputs when the number of measurements made is of order $O(n)$ Jozsa and Van den Nest (2014).

In this paper, we present two new efficient classical algorithms for approximately evaluating the output probabilities of Clifford circuits with nonstabilizer inputs. Our first algorithm shows that the output distribution of Clifford circuits with mixed product states can be efficiently approximated, with respect to the $l_{1}$ norm, for a large fraction of Clifford circuits. This algorithm explicitly reveals the role of mixedness of the input states in affecting the running time of the simulation, which decreases as the inputs become more mixed.

Our second algorithm shows that such an efficient approximation algorithm still exists in the case where the inputs are pure nonstabilizer states, as long as we impose a suitable restriction on the number of measured qubits. This restriction depends on a magic monotone called the Pauli rank that we introduce in this paper. This algorithm also explicitly links the simulation time to the amount of magic in the input states, and implies that for Clifford circuits with magic input states, it is possible in certain cases to achieve an efficient classical approximation of the output probability even when $O(n)$ qubits are measured. This is in contrast to the hardness result in Jozsa and Van den Nest (2014), which shows that sampling from those output probabilities is hard. Finally, we apply our results to give an efficient approximation algorithm for some restricted quantum computation models, like Clifford circuits with solely magic state inputs (CM), Pauli-based computation (PBC) and instantaneous quantum polynomial time (IQP) circuits.

II Main results

Let $P^{n}$ be the set of all Hermitian Pauli operators on $n$ qubits, i.e., operators that can be written as the $n$-fold tensor product of the single-qubit Pauli operators $\set{I,X,Y,Z}$ with sign $\pm 1$. The Clifford unitaries on $n$ qubits are the unitaries that maps Pauli operators to Pauli operators, that is, $\mathcal{C}l_{n}=\set{U\in U(2^{n}):UPU^{\dagger}\in P^{n},\forall P\in P^{n}}$. Stabilizer states are pure states of the form $U\ket{0}^{\otimes n}$ Aaronson and Gottesman (2004), where $U$ is some Clifford unitary.

Here, we consider Clifford circuits with product input states $|0\rangle\!\langle 0|^{\otimes n}\otimes^{m}_{i=1}\rho_{i}$, and measurements on $k$ qubits. If either $m$ or $k$ is $O(\log n)$, the output probabilities can be efficiently simulated classically by the Gottesman-Knill theorem Gottesman (1999); Jozsa and Van den Nest (2014). However, if both $m$ and $k$ are greater than $O(\log n)$, we show that the output probability of such circuits can still be approximated efficiently with respect to the $l_{1}$ norm for a large fraction of Clifford circuits.

II.1 Mixed input states

We first consider the case where all $\rho_{i}$ are mixed states and give an efficient classical algorithm to approximate the output probabilities.

Theorem 1**.**

Given a Clifford circuit $\mathcal{C}$ on $n+m$ qubits with input state $|0\rangle\!\langle 0|^{\otimes n}\otimes^{m}_{i=1}\rho_{i}$ and measurement on each qubit in the computational basis, there exists a classical algorithm to approximate the output probabilities of the circuit up to $l_{1}$ norm $\delta$ in time $(n+m)^{O(1)}m^{O(\log(\sqrt{\alpha}/\delta)/\lambda)}$ for at least $1-\frac{2}{\alpha}$ fraction of circuits $\mathcal{C}$, where $\lambda=\min\set{\lambda_{i}}_{i}$, with $\lambda_{i}=1-\sqrt{2\operatorname{Tr}\left[\rho^{2}_{i}\right]-1}$, is a measure of the mixedness of the input state $\rho_{i}$.

The proof of the Theorem is presented in Appendix A. The theorem shows that the efficiency of the classical simulation increases with the mixedness of the input states.

Next, we show that the result in Theorem 1 can be easily generalized to quantum circuits $\mathcal{C}$ which are slightly beyond Clifford circuits. To this end, we consider the Clifford hierarchy, a class of operations introduced by Gottesman and Chuang Gottesman and Chuang (1999) that has important applications in fault-tolerant quantum computation and teleportation-based state injection. Let $\mathcal{C}l^{(3)}_{n}$ be the third level of the Clifford Hierarchy, i.e., $\mathcal{C}l^{(3)}_{n}=\set{U\in U(2^{n}):UPU^{\dagger}\in\mathcal{C}l_{n},\forall P\in P^{n}}$. There are several important gates in the third level of Clifford Hierarchy, such as the $\pi/8$ gate (which we denote $T$) and the $CCZ$ gate Zeng et al. (2008). (Note that the set $\mathcal{C}l^{(3)}_{n}$ is not closed under multiplication. For example, $TH,T\in\mathcal{C}l^{(3)}_{n}$, but $THT\notin\mathcal{C}l^{(3)}_{n}$.) The following corollary shows that adding gates in $\mathcal{C}l^{(3)}$ to the circuits in Theorem 1 does not change (up to polynomial overhead) the efficiency of the classical simulation.

Corollary 2**.**

Let $\mathcal{C}=\mathcal{C}_{1}\circ V$ be a quantum circuit with input states $|0\rangle\!\langle 0|^{\otimes n}\otimes^{m}_{i=1}\rho_{i}$, where the gates in the circuit $\mathcal{C}_{1}$ are taken from the set of Clifford gates on $n+m$ qubits $\mathcal{C}l_{n+m}$ and $V$ is taken from the third level of Clifford hierarchy $\mathcal{C}l^{(3)}_{m}$ acting on $n+1,...,n+m$-th qubits. Assume that each each qubit is measured in the computational basis. Then, Theorem 1 stil holds if we replace $\mathcal{C}$ in Theorem 1 with $\mathcal{C}$ defined above.

The key property we use here is that the gates in the third level of the Clifford Hierarchy map Pauli operators to Clifford unitaries, which makes the proof of Theorem 1 still hold. (See a discussion of this in Appendix A. ) Although $\mathcal{C}l^{(3)}_{n}$ is not a group, the diagonal gates in $\mathcal{C}l^{(3)}_{n}$, denoted as $\mathcal{C}l^{(3)}_{n,d}$, forms a group Zeng et al. (2008); Cui et al. (2017). Since the $T$ gate and $CCZ$ gate both belong to $\mathcal{C}l^{(3)}_{n,d}$, the result in Theorem 1 still holds for the quantum circuits $\mathcal{C}=\mathcal{C}_{1}\circ\mathcal{C}_{2}$ where gates in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are chosen from $\mathcal{C}_{n+m}$ and $\mathcal{C}l^{(3)}_{m,d}$ respectively.

Since noise is inevitable in real physical experiments, it is important to consider the effects of noise in quantum computation. Recently, it has been demonstrated that if there is some noise on the random quantum gates Gao and Duan or measurements of IQP circuits Bremner et al. (2017), then there exists an efficient classical simulation of the output distribution of quantum circuits. In the rest of this subsection, we apply our results to two important subuniversal quantum circuits with noisy input states and give an efficient classical approximation algorithm for the output probabilities of the corresponding quantum circuits.

Example 1—First, we consider Clifford circuits with magic input states.

It is well known that the Clifford + $T$ gate set is universal for quantum computation. By magic state injection, circuits with this gate set can be efficiently simulated by Clifford circuits with magic state $\ket{T}$ inputs, where $\ket{T}=\frac{1}{\sqrt{2}}(\ket{0}+e^{i\pi/4}\ket{1})$. It has been shown that $\mathsf{postCM}=\mathsf{postBQP}$ Yoganathan et al. , and thus output probabilities are $\#\mathsf{P}$-hard approximate up to some constant relative error Kuperberg (2015); Fujii and Morimae (2017); Hangleiter et al. (2018). However, if there is some independent depolarizing error acting on each input magic state, e.g., the input state on each register is $(1-\epsilon)|T\rangle\!\langle T|+\epsilon\frac{I}{2}$, then Theorem 1 implies directly that there exists a classical algorithm to approximate the output probability up to $l_{1}$ norm $\delta$ in time $n^{O(\log(1/\delta)/\epsilon)}$ for a large fraction of the CM circuits with noisy inputs.

Example 2—IQP circuits have a simple structure with input states $\ket{0}^{\otimes n}$ and gates of the form $H^{\otimes n}DH^{\otimes n}$, where the diagonal gates in $D$ are chosen from the gate set $\set{Z,S,T,CZ}$.

It has been shown that $\mathsf{postIQP}=\mathsf{postBQP}$ Bremner et al. (2010) and thus, the output probabilities are $\#\mathsf{P}$-hard to approximate up to some constant relative error Kuperberg (2015); Fujii and Morimae (2017); Hangleiter et al. (2018). Also, if there is some depolarizing noise acting on each input state $\ket{0}$, i.e., each input state is a mixed state $(1-\epsilon)|0\rangle\!\langle 0|+\epsilon\frac{I}{2}$, then Theorem 1 implies that there exists a classical algorithm to approximate the output probability up to $l_{1}$ norm $\delta$ in time $n^{O(\log(1/\delta)/\epsilon)}$ for a large fraction of such IQP circuits. (The proof is presented in Appendix B in detail, which depends on the output distribution of IQP circuits in Appendix C. )

II.2 Pure nonstabilizer input states

As we can see, the running time in Theorem 1 blows up if the input state $\rho_{i}$ is pure. Here, we consider the case where all $\rho_{i}$ are pure nonstabilizer states, that is Clifford gates with the input state $\ket{0}^{\otimes n}\otimes^{m}_{i=1}\ket{\psi_{i}}$.

For pure states $\ket{\psi}$, the stabilizer fidelity Bravyi et al. is defined as follows

[TABLE]

where the maximization is taken over all stabilizer states. Here, we define

[TABLE]

It is easy to see that $\mu(\psi)=0$ iff $\ket{\psi}$ is a stabilizer state. Thus, $\mu$ quantifies the distance between a given state to the set of stabilizer states. Since each $\ket{\psi_{i}}$ is not a stabilizer state, it follows that $\mu(\psi_{i})>0$.

Next, let us introduce the Pauli rank for pure single qubit states $\ket{\psi}$. First, we write a pure state $\ket{\psi}$ in terms of its Bloch sphere representation $|\psi\rangle\!\langle\psi|=\frac{1}{2}\sum_{s,t\in\set{0,1}}\psi_{st}X^{s}Z^{t}$, where $\psi_{00}=1$ and $|\psi_{01}|^{2}+|\psi_{10}|^{2}+|\psi_{11}|^{2}=1$. We define the Pauli rank $\chi(\psi)$ to be the number of nonzero coefficients $\psi_{st}$. By the definition of Pauli rank, it is easy to see that $2\leq\chi(\psi)\leq 4$, and that $\ket{\psi}$ is a stabilizer state iff $\chi(\psi)=2$. Since each input state $\ket{\psi_{i}}$ is a nonstabilizer state, it follows that $\chi(\psi_{i})=3~{}\text{or}~{}4$. For example, for the magic state $\ket{T}$, the corresponding Pauli rank $\chi=3$. For $n$-qubit systems, the Pauli rank serves as a good candidate for a magic monotone as it is easier to compute than other magic monotones which require a minimization over all stabilizer states Bravyi and Gosset (2016); Howard and Campbell (2017); Veitch et al. (2014). (See a discussion of Pauli rank for $n$-qubit systems in Appendix D.)

Theorem 3**.**

Given a Clifford circuit $\mathcal{C}$ on $n+m$ qubits with input state $\ket{0}^{\otimes n}\otimes^{m}_{i=1}\ket{\psi_{i}}$ and measurements on $k$ qubits in the computational basis with $k\leq n+m-\sum^{m}_{i=1}\log_{2}(\chi(\psi_{i})/2)$ and $\chi(\psi_{i})$ being the Pauli rank of $\psi_{i}$, there exists a classical algorithm to approximate the output probability up to $l_{1}$ norm $\delta$ in time $(n+m)^{O(1)}m^{O(\log(\sqrt{\alpha}/\delta)/\mu)}$ for at least a $1-\frac{2}{\alpha}$ fraction of Clifford circuits $\mathcal{C}$, where $\mu:=\min_{i}\mu(\psi_{i})$ and $\mu(\psi_{i})$ is defined as (2).

The proof is presented in Appendix D. The maximal number of allowed measured qubits in this algorithm decreases with the amount of the magic in the input states, which is quantified by the Pauli rank. Curiously, the running time of this algorithm scales with the decrease in the amount of magic of the input states quantified by fidelity. This is contrary to the intuition that quantum circuits with more magic are harder to simulate. Similarly, if the quantum circuits are slightly beyond the Clifford circuits, for example, $\mathcal{C}=\mathcal{C}_{1}\circ V$ where the gates in $\mathcal{C}_{1}$ are Clifford gates in $\mathcal{C}l_{n+m}$ and $V$ is some unitary gate in the third level of the Clifford Hierarchy $\mathcal{C}l^{(3)}_{m}$, then the result in Theorem 3 still holds.

Combining Theorem 1 and 3, we have the following corollary for any product input state:

Corollary 4**.**

Let $\mathcal{C}$ be a Clifford circuit on $n+m_{1}+m_{2}$ qubits with input states $|0\rangle\!\langle 0|^{\otimes n}\otimes^{m_{1}}_{i=1}\rho_{i}\otimes^{m_{2}}_{j=1}|\psi_{j}\rangle\!\langle\psi_{j}|$, where each $\rho_{i}$ is a mixed state, and each $\ket{\psi_{j}}$ is a pure nonstabilizer state. Assume that measurements are performed on $k$ qubits in the computational basis, where $k\leq n+m_{1}+m_{2}-\sum^{m_{2}}_{j=1}\log_{2}(\chi(\psi_{i})/2)$ and $\chi(\psi_{i})$ is the Pauli rank of $\psi_{i}$. Then, there exists a classical algorithm to approximate the output probability with respect to the $l_{1}$ norm $\delta$ in time $(n+m_{1}+m_{2})^{O(1)}(m_{1}+m_{2})^{O(\log(\sqrt{\alpha}/\delta)/\epsilon)}$ for at least $1-\frac{2}{\alpha}$ fraction of Clifford circuits $\mathcal{C}$, where $\epsilon=\min\set{\lambda,\mu}$ and $\lambda:=\min_{i}\lambda_{i},\ \mu:=\min_{j}\mu(\psi_{j})$.

Now, let us apply our results to some restricted quantum computation models, such as Clifford circuits with solely magic state inputs (CM) and Pauli-based measurement (PBC), which gives an efficient simulation of $O(n)$ measurement with high probability.

Example 3—Theorem 3 implies the following result: for Clifford circuit $\mathcal{C}$ with input states $\ket{T}^{\otimes n}$ and measurement on $k$ qubits in computational basis with $k\leq(1-\log_{2}(3/2))n\approx 0.415n$, there exists a classical algorithm to approximate the output probability up to $l_{1}$ norm $\delta$ in time $n^{O((2+\sqrt{2})\log(\sqrt{\alpha}/\delta))}$ for at least $1-\frac{2}{\alpha}$ fraction of Clifford circuits $\mathcal{C}$, where $\mu(\ket{T})=1-\frac{1}{\sqrt{2}}$ and $\chi(\ket{T})=3$. This may be contrasted with the hardness result ruling out efficient classical sampling from this class of circuits Yoganathan et al. .

Example 4—A Pauli-Based Computation (PBC) is defined as a sequence of measurement of some Pauli operators $P_{i}\in P^{n}$, where the measurement outcome is $(-1)^{\sigma_{i}}$ with $\sigma_{i}\in\set{0,1}$ and the Pauli operators $\set{P_{i}}$ are commuting with each other. Here, the initial state is $\ket{T}$ (or $\ket{H}=\cos\frac{\pi}{8}\ket{0}+\sin\frac{\pi}{8}\ket{1}$, which is equivalent to $\ket{T}$ up to Clifford unitary Bravyi et al. (2016).). After $k$ steps, the probability of outcome $P(\sigma_{1},\ldots,\sigma_{k})=\bra{T^{\otimes n}}{\Pi}\ket{T^{\otimes n}}$, where $\Pi=2^{-k}\prod^{k}_{i=1}(I+(-1)^{\sigma_{i}}P_{i})$. Note that PBC was considered in the fault-tolerant implementation of quantum computation based on stabilizer codes, where the stabilizer codes provide a simple realization of nondestructive Pauli measurements Gottesman (1998); Steane (1997). Besides, it has been proved that the quantum computation based on Clifford+$T$ circuits can be simulated by PBC Bravyi et al. (2016). Thus, this implies that the output probability $P(\sigma_{1},\ldots,\sigma_{k})$ is $\#\mathsf{P}$-hard to simulate. It has been shown that any PBC on $n$ qubits can be classically simulated in $2^{cn}poly(n)$ time with $c\approx 0.94$ Bravyi et al. (2016). Here, Theorem 3 implies that if the measurement steps $k\leq(1-\log_{2}(3/2))n\approx 0.415n$, then there exists a classical algorithm to approximate the output probability up to $l_{1}$ norm $\delta$ in time $n^{O((2+\sqrt{2})\log(1/\delta))}$ for a large fraction of PBC.

III Conclusion

In this work, we investigated the problem of evaluating the output probabilities of Clifford circuits with nonstabilizer input states. First, we provided an efficient classical algorithm to approximate the output probability of the Clifford circuits with mixed input states and showed that the running time scales with the increase in the purity of input states. Second, we showed that a modification of this algorithm gives an efficient classical simulation for pure nonstabilizer states, under some restriction on the number of measured qubits that is determined by the Pauli rank of the input states. The Pauli rank we introduced in this work can be regarded as a good candidate for a magic monotone. We showed that these two results have several implications in other restricted quantum computation models such as Clifford circuits with magic input states, Pauli-based computation and IQP circuits.

Acknowledgements.

K. B. thanks Xun Gao for introducing the tensor network representation of quantum circuits to him, and for fruitful discussions related to this topic. K.B. acknowledges the Templeton Religion Trust for the partial support of this research under grants TRT0159 and Zhejiang University for the support of an Academic Award for Outstanding Doctoral Candidates. D.E.K. is funded by EPiQC, an NSF Expedition in Computing, under grant CCF-1729369.

Appendix A Proof of Theorem 1

A.1 Efficient evaluation of Fourier coefficients

First, let us define the Fourier transformation on a single qubit state, inspired by Gao and Duan . Given a single qubit state $\rho\in D(\mathbb{C}^{2})$, we can write it in terms of its Bloch sphere representation

[TABLE]

where $\rho_{00}=1$ and $|\rho_{10}|^{2}+|\rho_{01}|^{2}+|\rho_{11}|^{2}\leq 1$.

Given $a,b\in\mathbb{F}_{2}$, it is easy to verify that

[TABLE]

Thus, we can define the Fourier transformation on the state $\rho$ as follows

[TABLE]

Note that for $t=s=0$, the above Fourier transformation is equal to the completely depolarizing channel. And the equation (4) is the inverse Fourier transformation of (5).

Given the input states $|0\rangle\!\langle 0|^{\otimes n}\otimes^{m}_{i=1}\rho_{i}$ with Clifford unitary $U$, the output probability $q(\vec{y})$ is

[TABLE]

for any $\vec{y}\in\mathbb{F}^{n+m}_{2}$. Let us denote the Pauli operators $Z^{\vec{a}}:=\otimes^{m}_{i=1}Z^{a_{i}},X^{\vec{b}}:=\otimes^{m}_{i=1}X^{b_{i}}$ for any $\vec{a},\vec{b}\in\mathbb{F}^{m}_{2}$ to be operators acting on the latter $m$ qubits. Now, let us insert $X^{\vec{b}}Z^{\vec{a}}$ into the $m$ mixed states as follows

[TABLE]

Hence, the output probability $q(\vec{y})=q_{\vec{0},\vec{0}}(\vec{y})$. Then, let us take the Fourier transformation with respect to $\vec{a},\vec{b}$ and the corresponding Fourier coefficient is

[TABLE]

By equation (5), we have

[TABLE]

where $\rho^{(i)}_{s_{i}t_{i}}$ is the coefficient of $\rho_{i}$ in the corresponding Bloch sphere representation. Since $U$ is a Clifford unitary, then

[TABLE]

where the Pauli operators $P_{i}:=UZ_{i}U^{\dagger}$ for $1\leq i\leq n$ and $P_{j}:=UX^{s_{j}}Z^{t_{j}}U^{\dagger}$ for $1\leq j\leq m$ and they are commuting with each other. Thus, by Gottesman-Knill Theorem, the Fourier coefficients $\hat{q}_{\vec{s},\vec{t}}$ can be evaluated in classical $O((n+m)^{3})$ time .

A.2 Exponential decay of Fourier coefficients

Since $\rho$ is a mixed state in $D(\mathbb{C}^{2})$, it can always be written as $\rho=(1-\lambda)\sigma+\frac{\lambda}{2}I$, where $\sigma$ is a pure state and $\lambda=1-\sqrt{2\operatorname{Tr}\left[\rho^{2}\right]-1}$. The pure state $\sigma$ also has the Bloch sphere representation

[TABLE]

where $\sigma_{00}=1$ and $|\sigma_{10}|^{2}+|\sigma_{01}|^{2}+|\sigma_{11}|^{2}=1$. We have the following relationship between the coefficients $\rho_{st}$ and $\sigma_{st}$ for any $s,t\in\mathbb{F}_{2}$.

Lemma 5**.**

Given a mixed state $\rho=(1-\lambda)\sigma+\frac{\lambda}{2}I$, where $\rho,\sigma$ has Bloch sphere representation given by (3) and (9) respectively, then we have

[TABLE]

for any $s,t\in\mathbb{F}_{2}$, where $w(s,t)$ is defined as

[TABLE]

Proof.

This is because

[TABLE]

where $w(s,t)$ is defined as (11).

∎

Each mixed input state $\rho_{i}$ can be written as $\rho_{i}=(1-\lambda_{i})\sigma_{i}+\frac{\lambda_{i}}{2}I$ where $\sigma_{i}$ is a pure state. Consider the quantum circuit with input state $|0\rangle\!\langle 0|^{\otimes n}\otimes^{m}_{i=1}\sigma_{i}$ and Clifford unitary $U$, the output probability $p(\vec{y})$ is equal to

[TABLE]

Similar to $q(\vec{y})$, we insert $X^{\vec{b}}Z^{\vec{a}}$ into the circuit and define $p_{\vec{a},\vec{b}}$ as follows

[TABLE]

Then the corresponding Fourier coefficient can also be expressed as follows,

[TABLE]

where $\sigma^{(i)}_{s_{i}t_{i}}$ is the coefficient of $\sigma_{i}$ in the corresponding Bloch sphere representation. By Lemma 5, it is easy to see that

[TABLE]

where $\lambda=\min_{i}\lambda_{i}$ and $w(\vec{s},\vec{t})$ is defined as

[TABLE]

A.3 Good approximation with respect to $l_{1}$ norm

The following lemma regarding Clifford unitaries on $n$ qubits is necessary the proof,

Lemma 6** (Dankert et al. (2009)).**

The uniform distribution of Clifford unitaries on $n$ qubits is an exact 2-design, that is, for any $A,B,W$, we have

[TABLE]

where $\mathbb{E}_{U\sim\mathcal{C}l_{n}}:=\frac{1}{|\mathcal{C}l_{n}|}\sum_{U\sim\mathcal{C}l_{n}}$ and

[TABLE]

Now, let us prove Theorem 1. Let us define

[TABLE]

which gives an family of unnormalized probability distribution $\set{q^{\prime}_{\vec{a},\vec{b}}}$as $q^{\prime}_{\vec{a},\vec{b}}(\vec{y})=\sum_{\vec{s},\vec{t}}\hat{q}^{\prime}_{\vec{s},\vec{t}}(\vec{y})(-1)^{\vec{s}\cdot\vec{a}+\vec{t}\cdot\vec{b}}$ for each output $\vec{y}\in\mathbb{F}^{n+m}_{2}$ Then we show that $q^{\prime}_{\vec{0},\vec{0}}(\vec{y})$ gives a good approximation of $q_{\vec{0},\vec{0}}(\vec{y})$ with respect to $l_{1}$ norm

[TABLE]

for a large fraction of Clifford circuits. First, since $\hat{q}_{\vec{s},\vec{t}}(\vec{y})$ depends on the Clifford unitaries $U$, denote it as $\hat{q}_{\vec{s},\vec{t}}(\vec{y})[U]$, then it is easy to show that

[TABLE]

where $U^{\prime}=U\circ Z^{\vec{a}}X^{\vec{b}}$ is also a Clifford unitary for any $\vec{a},\vec{b}\in\mathbb{F}^{m}_{2}$ and $Z^{\vec{a}}X^{\vec{b}}$ act on the $n+1,\ldots,n+m$th qubits. Thus

[TABLE]

Moreover,

[TABLE]

where the first line comes from the Cauchy-Schwarz inequality, the third line comes from the Parseval identity, and the fourth line comes from the fact that $|\hat{q}_{\vec{s},\vec{t}}(\vec{y})|\leq(1-\lambda)^{w(\vec{s},\vec{t})}|\hat{p}_{\vec{s},\vec{t}}(\vec{y})|$. According to Lemma 6, we have

[TABLE]

Thus

[TABLE]

By Markov’s inequality, we have

[TABLE]

Therefore, to obtain the $l_{1}$ norm up to $\delta$, we need take $l=O(\log(\sqrt{\alpha}/\delta)/\lambda)$ and evaluate the Fourier coefficients $\hat{q}^{\prime}_{\vec{s},\vec{t}}(\vec{y})$ with $w(\vec{s},\vec{t})\leq l$, where total amount of such Fourier coefficients is $\sum_{i\leq l}3^{i}C^{i}_{m}\leq 3^{l}m^{l}$. Thus, there exists a classical algorithm to approximate each output probability $q(\vec{y})$ in time $O((n+m)^{3})m^{l}=(n+m)^{O(1)}m^{O(\log(\sqrt{\alpha}/\delta)/\lambda)}$ with $l_{1}$ norm less than $\delta$ for at least $1-\frac{2}{\alpha}$ fraction of Clifford circuits. Thus, we finish the proof of Theorem 1.

A.4 Slightly beyond Clifford circuits

Now, let us consider the quantum circuit $\mathcal{C}=\mathcal{C}_{1}\circ V$ with input state $|0\rangle\!\langle 0|^{\otimes n}\otimes^{m}_{i=1}\rho_{i}$ and the gates in circuits $\mathcal{C}_{1}$ taken from the set of Clifford gates on $n$ qubits $\mathcal{C}l_{n+m}$ and $V$ is taken from the third level of Clifford hierarchy $\mathcal{C}l^{(3)}_{m}$ acting on $n+1,\ldots,(n+m)$th qubits. The proof of Corollary 2 is almost the same as that of Theorem 1. We only need to show the corresponding Fourier coefficients of $q_{\vec{a},\vec{b}}$ also can be evaluated in $O((n+m)^{3})$ time, where

[TABLE]

and $V\in\mathcal{C}l^{(3)}_{m},U\in\mathcal{C}l_{n+m}$. Then the Fourier coefficient $\hat{q}_{\vec{s},\vec{t}}(\vec{y})$ is equal to

[TABLE]

Since $V\in\mathcal{C}l^{(3)}_{m}$, then $V\otimes^{m}_{i=1}X^{s_{i}}Z^{t_{i}}V^{\dagger}\in\mathcal{C}l_{m}$. Thus,

[TABLE]

where $U,U^{\prime}=V\otimes^{m}_{i=1}X^{s_{i}}Z^{t_{i}}V^{\dagger}U^{\dagger}$ are both Clifford unitaries. Thus, the Fourier coefficient $\hat{q}_{\vec{s},\vec{t}}$ can also be evaluated in $O((n+m)^{3})$ time by Gottesman-Knill Theorem. Therefore, it is easy to prove Corollary 2 by following the proof of Theorem 1.

Appendix B Efficient classical simualtion of IQP circuits with noisy input states

In this section, we will prove the following proposition in Example 2:

Proposition 7**.**

Given an IQP circuit $H^{\otimes n}DH^{\otimes n}$ with the diagonal unitaries chosen from the gate set $\set{CZ,Z,S,T}$, if there is depolarizing nosie acting on each input state, i.e., input state is $((1-\epsilon)|0\rangle\!\langle 0|+\frac{\epsilon}{2}I)^{\otimes n}$, then there exists an efficient classical algorithm to approximate the output probabilities up to $l_{1}$ norm $\delta$ in time $n^{O(\log(\sqrt{\alpha}/\delta)/\epsilon)}$ for at least $1-\frac{2}{\alpha}$ fraction of IQP circuits.

Proof.

The proof is similar to that of Theorem 1. If the state $\rho$ has some specific form as $\rho=\frac{1}{2}(\rho_{0}I+\rho_{1}Z)$, then we can simplify the Fourier transformation (5) as

[TABLE]

Given an IQP circuit $H^{\otimes n}DH^{\otimes}$ with noisy input states $\rho^{\otimes n}$, $\rho=(1-\epsilon)|0\rangle\!\langle 0|+\epsilon\frac{I}{2}$, and gates in $D$ chosen from the gate set $\set{CZ,Z,S,T}$, then the output probability $q(\vec{y})$ is equal to

[TABLE]

Similar to the proof of Theorem 1, we insert $X^{\vec{a}}$ into the circuits for any $\vec{a}\in\mathbb{F}^{n}_{2}$ and define $q_{\vec{a}}(\vec{y})$ as follows

[TABLE]

Then let us take the Fourier transformation with respect to $\vec{a}$ and the corresponding Fourier coefficient is

[TABLE]

where the second last equality comes from (25).

Besides,

[TABLE]

where the diagonal part $D$ can be written as $D^{\prime}\circ\otimes^{n}_{i=1}T^{\gamma_{i}}$ with $\gamma_{i}\in\mathbb{F}_{2}$ and the gates in $D^{\prime}$ chosen from the gate set $\set{CZ,Z,S}$. It is easy to verify that

[TABLE]

for any $\gamma_{i},s_{i}\in\set{0,1}$. That is, $DH^{\otimes n}\otimes_{i}Z^{s_{i}}H^{\otimes n}D^{\dagger}$ is a Clifford circuit. Thus, each Fourier coefficient can be evaluated in $O(n^{3})$ by Gottesman-Knill Theorem.

We also consider the same IQP circuits with input states $|0\rangle\!\langle 0|^{\otimes n}$, then output probability $p(\vec{y})=\bra{\vec{y}}H^{\otimes n}DH^{\otimes n}|0\rangle\!\langle 0|^{\otimes n}H^{\otimes n}DH^{\otimes n}\ket{\vec{y}}$. Similarly, we insert the operator $X^{\vec{a}}$ as follows

[TABLE]

And the corresponding Fourier coefficient is

[TABLE]

Comparing (28) with (32), we have the following relation

[TABLE]

where $|\vec{s}|=\sum_{i}s_{i}$ is the Hamming weight of $\vec{s}\in\mathbb{F}^{n}_{2}$.

Let us define

[TABLE]

which gives an family of unnormalized probability distribution $\set{q^{\prime}_{\vec{a}}}$as $q^{\prime}_{\vec{a}}(\vec{y})=\sum_{\vec{s}}\hat{q}^{\prime}_{\vec{s}}(\vec{y})(-1)^{\vec{s}\cdot\vec{a}}$ for each output $\vec{y}\in\mathbb{F}^{n}_{2}$. Then we will show that $q^{\prime}_{\vec{0}}(\vec{y})$ gives a good approximation of $q_{\vec{0}}(\vec{y})$ with respect to $l_{1}$ norm

[TABLE]

for a large fraction of IQP circuits. We denote $\mathbb{D}_{n}$ to be the set of of diagonal part of IQP circuits where the diagonal gates are chosen from $\set{CZ,Z,S,T}$. Since $\hat{q}_{\vec{s}}(\vec{y})$ depends on the IQP circuits, denote it as $\hat{q}_{\vec{s},\vec{t}}(\vec{y})[D]$, then it is easy to verify that

[TABLE]

where $D^{\prime}=D\circ Z^{\vec{a}}$ also belongs to $\mathbb{D}_{n}$. Thus

[TABLE]

And

[TABLE]

where the first line comes from the Cauchy-Schwarz inequality, the third line comes from Parvesal identity, and the fourth line comes from the fact that $\hat{q}_{\vec{s}}(\vec{y})=(1-\epsilon)^{|\vec{s}|}\hat{p}_{\vec{s}}(\vec{y})$. According to Lemma 8 in Appendix C, we have

[TABLE]

Thus, we have

[TABLE]

Therefore, by Markov’s inequality, we have

[TABLE]

Therefore, to obtain the $l_{1}$ norm up to $\delta$, we need take $l=O(\log(\sqrt{\alpha}/\delta)/\epsilon)$ and the total computational complexity is $O(n^{3}n^{l})=n^{O(\log(\sqrt{\alpha}/\delta)/\epsilon)}$. ∎

Appendix C Distribution of IQP circuits based on Gowers uniformity norm

Here we consider IQP circuits, which can be represented by $H^{\otimes n}DH^{\otimes n}\ket{0}^{\otimes n}$, where the gates in the diagonal part $D$ are chosen from the gate set $\set{CZ,Z,S,T}$. Then the output distribution is $p(\vec{y})=|\bra{\vec{y}}H^{\otimes n}DH^{\otimes n}\ket{0}^{\otimes n}|^{2}=|\hat{f}(\vec{y})|^{2}$ for any $\vec{y}\in\mathbb{F}^{n}_{2}$, where $\hat{f}(\vec{y})=\frac{1}{2^{n}}\sum_{\vec{x}\in\mathbb{F}^{n}_{2}}f(\vec{x})(-1)^{\vec{y}\cdot\vec{x}}$ and the function $f$ can be expressed as

[TABLE]

where $\alpha_{ij},\beta_{i},\gamma_{i},t_{i}\in\mathbb{F}_{2}$, denote the number of $CZ$ between $i$th and $j$th qubits, Z gate on $i$th qubit, S gate on $i$th gate and T gate on $i$th gate. Since $T^{2}=S,S^{2}=Z$ and $Z^{2}=I$, then there are at most one $T$, $S$, $Z$ gate on each qubit respectively. Thus, $\vec{\beta},\vec{\gamma},\vec{t}\in\mathbb{F}^{n}_{2}$ and the Hamming weight $|\vec{\beta}|,|\vec{\gamma}|,|\vec{t}|$ is the number of $Z$, $S$ and $T$ gates in the IQP circiut.

In fact, the function $f$ can be rewritten as follows

[TABLE]

where $A_{ii}=\gamma_{i}$ and $A_{ij}=A_{ji}=\alpha_{ij}$ for $i\neq j$. That is, the matrix $A$ is a symmetric $0-1$ matrix.

Now, let us introduce the Gowers uniformity norm here. Let $G$ be a finite additive group and $f:G\to\mathbb{C}$ and an integer $d\geq 1$. Then the Gowers uniformity norm $\left\lVert f\right\rVert_{U^{d}(G)}$ Tao and Vu (2006) is defined as

[TABLE]

where $\Delta_{h}f(x):=f(x+h)\overline{f(x)}$. Here we take $G=\mathbb{F}^{n}_{2}$ and the Fourier transformation for $f:\mathbb{F}^{n}_{2}\to\mathbb{C}$ is defined as $\hat{f}(\vec{y})=\mathbb{E}_{\vec{x}\in\mathbb{F}^{n}_{2}}f(x)(-1)^{\vec{x}\cdot\vec{y}}$, where $\mathbb{E}_{\vec{x}\in\mathbb{F}^{n}_{2}}:=\frac{1}{2^{n}}\sum_{\vec{x}\in\mathbb{F}^{n}_{2}}$. One important property of Gowers uniformity norm, which we will use in the following section to demonstrate the distribution of IQP circuits, is the following equality Tao and Vu (2006)

[TABLE]

For IQP circuits with diagonal gates chosen from $\set{CZ,Z,CCZ}$ randomly, it has been proved that the average value of the second moment of output probability satisfies that $\sum_{\vec{y}}p^{2}_{D}(\vec{y})\leq\alpha 2^{-n}$, where $\alpha$ is some constant Bremner et al. (2016). Here, we consider the case where the gates in the diagonal part $D$ are chosen uniformly, i.e., $P(\alpha_{ij=1})=P(\beta_{i}=1)=P(\gamma_{i}=1)=P(t_{i}=1)=1/2$, then we can give the exact value of average value of the second moment of the output probability of random IQP circuits.

Lemma 8**.**

Given an IQP circuit, if the gates in the diagonal part $D$ can be chosen uniformly, then

[TABLE]

Proof.

Due to the equation (38), we have

[TABLE]

For the function $f(\vec{x})=(-1)^{\vec{\beta}\cdot\vec{x}}i^{\vec{x}A\vec{x}}e^{i\pi/4\vec{t}\cdot\vec{x}}$, the Gowers uniformity norm $\left\lVert f\right\rVert_{U^{2}(\mathbb{F}^{n}_{2})}$ can be expressed as follows

[TABLE]

It is easy to verify that

[TABLE]

for any $t,a,b\in\mathbb{F}_{2}$. Thus, we have

[TABLE]

The expected value of $\sum_{\vec{y}\in\mathbb{F}^{n}_{2}}p^{2}_{D}(\vec{y})$ over the random IQP circuits is

[TABLE]

Since

[TABLE]

then the above equation is equal to

[TABLE]

where the equality comes from the fact that

[TABLE]

when $(a_{i},b_{i}),(a_{j},b_{j})$ are chosen from $\set{(0,0),(0,1),(1,0)}$. Moreover, for $(a_{i},b_{i}),(a_{j},b_{j})\in\set{(0,0),(0,1),(1,0)}$, we have

[TABLE]

Thus,

[TABLE]

Therefore, we obtain the result that

[TABLE]

∎

Besides, based on the Gowers uniformity norm, we can also give an approximation of the second moment for any IQP circuit.

Proposition 9**.**

Given an IQP circuit with the diagonal gates chosen from $\set{CZ,Z,S,T}$, then the output probability of this circuit satisfies the following property,

[TABLE]

where the constant $c=\log\frac{4}{3}>0$, $A(\vec{t})$ is the matrix obtained from $A$ by removing the rows and columns $i$ such that $t_{i}=1$ and $\mathrm{Rank}(A(\vec{t}))$ denotes the rank of the matrix $A(\vec{t})$ in $\mathbb{F}_{2}$. Moreover, if $\vec{t}=0$, i.e., there is no $T$ gate, then

[TABLE]

Proof.

Due to the equation (38) and Lemma 8, we have

[TABLE]

Thus, we need estimate the Gower uniform norm $\left\lVert f\right\rVert_{U^{2}(\mathbb{F}^{n}_{2})}$ for the phase polynomial $f(\vec{x})=(-1)^{\vec{\beta}\cdot\vec{x}}i^{\vec{x}A\vec{x}}e^{i\pi/4\vec{t}\cdot\vec{x}}$ by the Hamming weight $|\vec{t}|$ and the rank of the symmetric matrix $A$.

Without loss of generality, we assume the first $k=|\vec{t}|$ qubits have $T$ gates, i.e., $t_{1}=\ldots=t_{k}=1$, and the remaining qubits do not have $T$ gate, then we can decompose the symmetric matrix A as follows

[TABLE]

where $A_{k,k}$ is an $k\times k$ symmetric matrix, $A_{n-k,n-k}$ is an $(n-k)\times(n-k)$ symmetric matrix and $A_{n-k,k}=A^{t}_{k,n-k}$. Similarly, we also decompose the vectors $\vec{a},\vec{b}$ as

[TABLE]

where $\vec{a}_{k},\vec{b}_{k}\in\mathbb{F}^{k}_{2}$ and $\vec{a}_{n-k},\vec{b}_{n-k}\in\mathbb{F}^{n-k}_{2}$. Thus,

[TABLE]

Since

[TABLE]

then

[TABLE]

Besides, for any $\vec{x},\vec{y}\in\mathbb{F}^{n-k}_{2}$,

[TABLE]

where $\text{Rank}(A_{n-k,n-k})$ denotes the rank of the matrix $A_{n-k,n-k}$ in $\mathbb{F}_{2}$. Therefore,

[TABLE]

where $c=\log\frac{4}{3}$.

Moreover, if $\vec{t}=0$, then

[TABLE]

∎

Appendix D Efficent classical simulation with pure nonstabilizer input states

D.1 Proof of Theorem 3

Lemma 10**.**

For any pure state $\ket{\psi}$ in $D(\mathbb{C}^{2})$, the stabilizer fidelity can be expressed as

[TABLE]

Proof.

This follows directly from the fact the single-qubit stabilizer states are the eigenstates of $X,Y,Z$, that is, the stabilizer states have the form $|\phi\rangle\!\langle\phi|=\frac{I\pm P}{2}$, where $P\in\set{X,Y,Z}$. ∎

Thus $\mu(\psi)$ can also be expressed as

[TABLE]

Now, let us begin the proof of Theorem 3. Since $\ket{\psi}$ has the Bloch sphere representation as $|\psi\rangle\!\langle\psi|=\frac{1}{2}\sum_{s,t\in\mathbb{F}_{2}}\psi_{st}X^{s}Z^{t}$, it is easy to see that

[TABLE]

for any $(s,t)\neq(0,0)$.

Without loss of generality, we assume the first $k$ qubits are measured as the swap gate belongs to $\mathcal{C}l_{n+m}$. Then the output probability is

[TABLE]

for any $\vec{y}\in\mathbb{F}^{k}_{2}$, where $I_{n+m-k}$ denotes the identity on the $k+1,...,(n+m)$th qubits. Let us insert the Pauli operator $X^{\vec{b}}Z^{\vec{a}}$ into the circuit and the corresponding output probability

[TABLE]

The corresponding Fourier coefficient is

[TABLE]

Now let us define the reference Hermitian operator with respect to $\psi$ as follows

[TABLE]

where the function $\mathrm{sgn}$ is defined as $\mathrm{sgn}(x)=1$ if $x>0$, $\mathrm{sgn}(x)=0$ if $x=0$. It is easy to verify that $\operatorname{Tr}\left[O(\psi)\right]=1,\operatorname{Tr}\left[O(\psi)^{2}\right]=\frac{\chi(\psi)}{2}$, where $\chi(\psi)$ is the Pauli rank of $\psi$. Besides, we have

[TABLE]

Combined with (47), we have the following relation

[TABLE]

for any $s,t\in\mathbb{F}_{2}$. We also define $o_{\vec{a},\vec{b}}$ as follows

[TABLE]

where each $O_{i}$ is the reference Hermitian operator with respect to $\psi_{i}$ defined as (51) and the corresponding Fourier coefficient is

[TABLE]

Thus, in terms of the relation (53), we have

[TABLE]

where $\mu=\min_{i}\mu(\psi_{i})$ and $w(\vec{s},\vec{t})$ is defined as (16).

Let us define

[TABLE]

which gives a family of unnormalized probability distribution $\set{q^{\prime}_{\vec{a},\vec{b}}}$as $q^{\prime}_{\vec{a},\vec{b}}(\vec{y})=\sum_{\vec{s},\vec{t}}\hat{q}^{\prime}_{\vec{s},\vec{t}}(\vec{y})(-1)^{\vec{s}\cdot\vec{a}+\vec{t}\cdot\vec{b}}$ for each output $\vec{y}\in\mathbb{F}^{k}_{2}$ Similar to the proof of Theorem 1, we show that $q^{\prime}_{\vec{0},\vec{0}}(\vec{y})$ gives a good approximation of $q_{\vec{0},\vec{0}}(\vec{y})$ with respect to $l_{1}$ norm for a large fraction of Clifford circuits.

It is easy to verify that the equations (20) and (21) still hold, and we can repeat the process of inequality (22) and obtain the following inequality

[TABLE]

By the Lemma 6, we have

[TABLE]

Since $k\leq n+m-\sum^{m}_{i=1}\log_{2}\left(\frac{\chi(\psi_{i})}{2}\right)$ , then we have

[TABLE]

By Markov’s inequality, we have

[TABLE]

Therefore, to obtain the $l_{1}$ norm up to $\delta$, we need take $l=O(\log(\sqrt{\alpha}/\delta)/\mu)$ and evaluate the Fourier coefficients $\hat{q}^{\prime}_{\vec{s},\vec{t}}(\vec{y})$ with $w(\vec{s},\vec{t})\leq l$, where the total amount of such Fourier coefficients is $\sum_{i\leq l}3^{i}C^{i}_{m}\leq 3^{l}m^{l}$. Thus, there exists a classical algorithm to approximate each output probability $q(\vec{y})$ in time $O((n+m)^{3})m^{l}=(n+m)^{O(1)}m^{O(\log(\sqrt{\alpha}/\delta)/\mu)}$ with $l_{1}$ norm less than $\delta$ for at least $1-\frac{2}{\alpha}$ fraction of Clifford circuits. Thus, we finish the proof of Theorem 3.

Moreover, if the quantum circuit $\mathcal{C}$ is slightly beyond the Clifford circuits, e.g. $\mathcal{C}=\mathcal{C}_{1}\circ V$ where the gates in $\mathcal{C}_{1}$ are Clifford gates and $V$ is some unitary gate in third level of Clifford Hierarchy, then the result in Theorem 3 still works, as the unitary in third level of Clifford hierarchy maps Pauli operators to Clifford unitaies and thus the discussion in Appendix A.4 still works.

D.2 Property of Pauli rank

At the end of this section, let us introduce several basic properties of Pauli rank. For any pure state $\ket{\psi}$ on $n$ qubits, we have the Bloch sphere representation

[TABLE]

where $\psi_{\vec{0},\vec{0}}=1$ and $\sum_{(\vec{s},\vec{t})\neq(\vec{0},\vec{0})}|\psi_{\vec{s},\vec{t}}|^{2}=2^{n}-1$. The Pauli rank is defined as the number of nonvanishing coefficients $\psi_{\vec{s},\vec{t}}$, that is,

[TABLE]

Then we have the following property for the Pauli rank.

Proposition 11**.**

Given an $n$-qubit pure state $\ket{\psi}$, we have

(i) $2^{n}\leq\chi(\psi)\leq 4^{n}$, $\chi(\psi)=2^{n}$ iff $\psi$ is a stabilizer state.

(ii) $\chi(\psi_{1}\otimes\psi_{2})=\chi(\psi_{1})\chi(\psi_{2})$.

Proof.

(i) $2^{n}\leq\chi(\psi)\leq 4^{n}$ follows directly from the definition. We only need prove $\chi(\psi)=2^{n}$ iff $\psi$ is a stabilizer state. In the backward direction, if $\psi$ is a stabilizer state, then it can be written as $|\psi\rangle\!\langle\psi|=\prod^{n}_{i=1}\frac{I+P_{i}}{2}$, where $P_{i}\in P^{n}$ and $P_{i}$ are commuting with each other. Thus, the Pauli rank of $\ket{\psi}$ is $2^{n}$. In the forward direction, if $\chi(\psi)=2^{n}$, then it can be represented as $|\psi\rangle\!\langle\psi|=\frac{1}{2^{n}}\sum^{2^{n}}_{i=1}P_{i}$ where $P_{1}=I$, each $P_{i}\in P^{n}$, and $P_{i},P_{j}$ are not equivalent in the sense that $\operatorname{Tr}\left[P_{i}P_{j}\right]=0$ for any $i\neq j$. First, we show that $P_{i}P_{j}=P_{j}P_{i}$ for any $i,j$. Otherwise, there exists $i_{0},j_{0}$ such that $P_{i_{0}}P_{j_{0}}=-P_{j_{0}}P_{i_{0}}$. Since $\psi$ is a pure state, then

[TABLE]

where the third inequality comes from the fact that $P_{i_{0}}P_{j_{0}}=-P_{j_{0}}P_{i_{0}}$. Since each $P_{i}P_{j}$ is equal to $i^{c_{ijk}}P_{k}$ for some $k$ and $n_{k}$ is the summation the these phases $i^{c_{ijk}}$, thus

[TABLE]

Then there is some $k_{0}$ such that $|n_{k_{0}}|\leq 2^{n}-1$, i.e., $\frac{|n_{k_{0}}|}{2^{n}}<1$, which contradicts with the representation of $\psi$. Thus, $P_{i}$ are commuting with each other. Next, we prove that this set of $\set{P_{i}}^{2^{n}}_{i=1}$ can be generated by some subset $S$ up to $\pm$ sign. For any $P_{i}$ not equal to identity, e.g., $P_{2}$, then there exists $U_{1}\in\mathcal{C}l_{n}$ such that $U_{1}P_{2}U^{\dagger}_{1}=Z\otimes I_{n-1}$, and for any $i$, $U_{1}P_{i}U^{\dagger}_{1}$ must have the form $Z^{a_{i}}\otimes P_{i,n-1}$, where $P_{i,n-1}\in P^{n-1}$ and they are commuting with each other. The generating set $S=\set{Z\otimes I_{n-1}}$. For some $P_{i,n-1}$ not equal to identity, e.g., $Z^{a_{3}}\otimes P_{3,n-1}$, there exists $U_{2}\in\mathcal{C}l_{n-1}$ such that $U_{2}U_{1}P_{3}U^{\dagger}_{1}U^{\dagger}_{2}=Z^{a_{3}}\otimes Z\otimes I_{n-2}$, and $U_{2}U_{1}P_{i}U^{\dagger}_{1}U^{\dagger}_{2}=Z^{a_{i}}\otimes Z\otimes P_{i,n-2}$. Then the generating set is updated to $S=\set{Z\otimes I_{n-1},Z^{a_{3}}\otimes Z\otimes I_{n-2}}$. Let us repeat the above process for another $n-2$ times, finially we will get some generating set $S=\set{g_{i}}^{n}_{i=1}$, where $g_{i}=Z^{c_{i,1}}\otimes\ldots\otimes Z^{c_{i,i-1}}\otimes Z\otimes I_{n-i}$. Moreover, the remaining Pauli operators must have the form $\pm\otimes^{n}_{i=1}Z^{a_{i}}$, which can be generated by the generating set $S$ up to $\pm$ sign. That is, there is a Clifford unitary map $U$ that maps $|\psi\rangle\!\langle\psi|$ to another pure state $|\psi^{\prime}\rangle\!\langle\psi^{\prime}|=\frac{1}{2^{n}}\sum_{\vec{a}\in\mathbb{F}^{n}_{2}}c_{|\vec{a}|}Z^{\vec{a}}$ where $c_{|\vec{a}|}=\pm 1$, $|\vec{a}|:=\sum_{i}a_{i}2^{i-1}$ and $c_{0}=1$. Repeating the argument (59) and (60) for the pure state $|\psi^{\prime}\rangle\!\langle\psi^{\prime}|$, we have $c_{|\vec{a}|}=\prod^{n}_{i=1}c^{a_{i}}_{2^{i-1}}$. Thus $|\psi^{\prime}\rangle\!\langle\psi^{\prime}|=\prod^{n}_{i=1}\frac{I+c_{2^{i-1}}Z_{i}}{2}$ where $Z_{i}$ denotes the Pauli $Z$ operator acting on the $i$th qubit. Therefore $\psi$ is a stabilizer state.

(ii) This property follows directly from the definition.

∎

Based on the above proposition and the fact that the Pauli rank is invariant under conjugation by Clifford unitaries, it is easy to see that the Pauli rank is a good candidate to quantify the magic in a state. Here, using the Pauli rank as a magic monotone is advantageous because it is easier to compute than previous magic monotones Bravyi and Gosset (2016); Howard and Campbell (2017); Veitch et al. (2014), which typically involve a minimization over all stabilizer states.