Quantum certification of state set and unitary channel

In the emerging quantum information technology, a fundamental task in building reliable quantum information processing devices is to obtain parameters of an unknown quantum state or device. The process is called tomography if all parameters are required to be known, which has been extensively studied since fifty years ago. Efficient quantum algorithms based on group representation theory have been proposed in [8, 16, 17]. In many scenarios, however, we are concerned only with whether the unknown state or operation satisfies specific property. For example, to assess the quality of a quantum chip after production, one needs only to check whether the circuit is close to a given unitary transformation, and it is unnecessary to get all parameters about this chip. This process is called certification, which usually saves samples and storage space compared with the quantum tomography. A survey on quantum certification is given in [14].

The abstract setting for certification can be described as follows. Given a known set $\mathcal{P}$ and an unknown element $x$, a tester (or an algorithm) $\mathcal{T}$ either accepts (i.e., reports $x\in\mathcal{P}$) or rejects (i.e., reports $x\notin\mathcal{P}_{\varepsilon}$) with some probability after measuring $x$, where $\mathcal{P}_{\varepsilon}:=\{y:\textnormal{dist}(y,\mathcal{P})\leq\varepsilon\}$ with dist denoting some metric. The tester $\mathcal{T}$ is eligible if the following conditions hold:

(1) (Completeness) If $x\in\mathcal{P}$, $\mathcal{T}$ accepts with probability at least 2/3;

(2) (Soundness) If $x\not\in\mathcal{P}_{\varepsilon}$, $\mathcal{T}$ accepts with probability at most 1/3.

The numbers $2/3,1/3$ have no special meaning and can be replaced by any constants $c>1/2,s<1/2$ respectively due to the confidence amplification by using repeating the test. If the tester accepts with certainty when $x\in\mathcal{P}$, we say the tester has perfect completeness.

Recall that the trace distance between two quantum states is $\textnormal{D}(\rho,\sigma):=\frac{1}{2}\|\rho-\sigma\|_{1}$, and their fidelity is $\textnormal{F}(\rho,\sigma):=\|\sqrt{\rho}\sqrt{\sigma}\|_{1}$. Given a known state $|{\varphi}\rangle$ and an unknown state $|{\psi}\rangle$, the aim is to decide whether $\psi=\varphi$, i.e., $|{\psi}\rangle=\textnormal{e}^{\textnormal{i}\theta}|{\varphi}\rangle$ for some real $\theta$, or $\textnormal{D}(\psi,\varphi)\geq\varepsilon$. In this task of certification, $\{\textnormal{e}^{\textnormal{i}\theta}|{\varphi}\rangle:\theta\in\mathbb{R}\}$ is the known set $\mathcal{P}$ and $|{\psi}\rangle$ is the unknown element $x$. It turns out $O(\varepsilon^{-2})$ copies of given states are sufficient for this task. The test is simply to perform the POVM $\{|{\varphi}\rangle\!\langle{\varphi}|,\mathds{1}-|{\varphi}\rangle\!\langle{\varphi}|\}$ on $n$ samples independently, and accept if and only if every outcome corresponds to the former POVM element. If $\textnormal{D}(\psi,\varphi)=\sqrt{1-\langle{\psi,\varphi}\rangle}\geq\varepsilon$, then $\langle{\psi,\varphi}\rangle^{n}\leq(1-\varepsilon^{2})^{n}\leq\frac{1}{3}$ and one can take $n=O(\varepsilon^{-2})$ for small $\varepsilon$.

In order to test whether a pair of unknown states $\rho$ and $\sigma$ on $\mathcal{H}$ are equal, the swap test [2] is usually used. The swap test is simply a two-outcome measurement $\{P,Q\}$, where $P$ and $Q$ are projectors onto the symmetric subspace and the antisymmetric subspace of $\mathcal{H}^{\otimes 2}$ respectively. The first outcome occurs with probability $\frac{1}{2}(1+\operatorname{tr}(\rho\sigma))$, and when this is the case, we say the swap test accepts or passes. By applying group theory, it is shown in [1] that $O(d/\varepsilon^{2})$ copies of quantum states suffices to distinguish whether an unknown $\rho$ is equal to some known $\sigma$ or $\varepsilon$-far from $\sigma$ in trace distance. This method is efficient since the tomography needs $\Omega(d^{2})$ copies of quantum states.

Let $\mathcal{P}$ be a finite set of pure states such that $\min_{\varphi\neq{\varphi^{\prime}}\in\mathcal{P}}\textnormal{D}(\varphi,\varphi^{\prime})=:\delta$, and let $\psi$ be an unkown pure state. Then $O(\max\{\varepsilon^{-2},\delta^{-2}\}\ln|\mathcal{P}|)$ copies suffice to distinguish whether $\psi\in\mathcal{P}$ or $\textnormal{D}(\psi,\mathcal{P})\geq\varepsilon$ [20]. Following the method in [1] we can use $O(\varepsilon^{-2}|\mathcal{P}|)$ copies of the unknown state $\rho$ to distinguish whether $\rho\in\mathcal{P}$ or $\min_{\sigma\in\mathcal{P}}\|\rho-\sigma\|_{2}\geq\varepsilon$. We also show that $O(\varepsilon^{-4}\ln|\mathcal{P}|)$ copies of unknown state $\psi$ suffice to distinguish whether $\psi\in\mathcal{P}$ or $\textnormal{D}(\psi,\mathcal{P})\geq\varepsilon$. When $\delta$ is small compared with $\varepsilon$, our method exhibits advantage by using notably less samples.

The certification of unitaries is quite different from that of quantum states in that the quantum unitary certification requires a double optimization, one is the input state for the quantum unitary and the other is the choice of measurement after the action of quantum unitary. The works in [5, 19, 18] used methods based on Monte Carlo sampling to estimate the fidelity of an unknown gate to a fixed one, and also studied the optimality of estimation strategy. Given an unknown unitary $U$ and a known or unknown unitary $V$, by using the Choi correspondence between quantum unitaries and states, there exists a tester that distinguishes whether their distance is zero or larger than $\varepsilon$ with $O(\varepsilon^{-2})$ uses of unitaries. By using Schur-Weyl decomposition, we show that for fixed dimension of the unitary, only $O(\varepsilon^{-1})$ uses of the unitaries suffice to achieve the same goal. Another advantage of our algorithm needs much fewer or no ancilla system compared with the method using Choi state.

Other properties of unitaries have been studied. Productness can be tested with $O(\varepsilon^{-2})$ copies [10] using the procedure first discussed in [13] which applies the swap test across each corresponding pair of subsystems of two copies of $|{\psi}\rangle$. Testing product unitaries can be reduced to testing product pure states using Choi isomorphism [10]. The tasks of testing Pauli matrices and Clifford gates are also studied in [15] and [12, 20] respectively.

The structure of the present paper is as follows. Section 2 gives a brief introduction to group representation theory which is a basic method for our analysis. Section 3 studies the certification of the membership of an unknown state in a known set. Section 4 studies the certification of the equality of two unitary quantum channels. Finally Section 5 gives a discussion.

In this section we give a brief introduction to important results in group representation theory and fix some notation, which will be used later. For more details on representation theory, we refer to [6, 7, 9]. Consider the following Schur-Weyl duality

where $\mathcal{H}_{\lambda}$ is the irreducible representation (irrep) of unitary group $\textnormal{U}(d)$ corresponding to partition $\lambda$ and $\mathcal{K}$ is the corresponding irrep of the symmetric group $S_{n}$. Here $\lambda\vdash(n,d)$, also written $\lambda\in\textnormal{Par}(n,d)$, means that $\lambda$ is a partition of $n$ with length at most $d$.

The dimension of $\mathcal{K}_{\lambda}$, for partition $\lambda$ of $n$, is given by

where $\widetilde{\lambda}_{i}:=\lambda_{i}+d-i$. It can be bounded by [11]

and furthermore

where $\bar{\lambda}:=\lambda/n$.

The character $\chi_{\lambda}^{L}$ for the irrep $\mathcal{H}_{\lambda}$ is

The dimension of $\mathcal{H}_{\lambda}$ is given by

which is bounded above by $(n+1)^{d(d-1)/2}$ [4].

In the study of testing whether an unknown state is close to or far from a given state and whether two unknown states are close to each other, Bădescu et al. [1] used the Chebyshev inequality to bound the deviation of a random variable from its expectation so that the completeness and soundness conditions can be satisfied. To be specific, for a real-valued random variable $X$ and a scalar $0<\gamma<\frac{1}{2}$, consider a tester which reports $\operatorname{\mathbb{E}}X\leq(1-2\gamma)\theta$ when $X\leq(1-\gamma)\theta$ is observed and reports $\operatorname{\mathbb{E}}X\geq\theta$ when $X>(1-\gamma)\theta$ is observed. When $\operatorname{\mathbb{E}}X\leq(1-2\gamma)\theta$, the probability that the tester reports correctly is

When $\operatorname{\mathbb{E}}X\geq\theta$, the probability that the tester reports correctly is

If $\textnormal{Var}X$ is small enough as compared with $\gamma\theta$, the tester is eligible. In our settings, let $X$ denote the random variable of measurement outcome of $M$ on a state $\varrho$. Obviously the expectation of $X$ is $\operatorname{\mathbb{E}}X=\operatorname{tr}(\varrho M)$ and the variance of $X$ is $\textnormal{Var}X=\operatorname{tr}(\varrho M^{2})-(\operatorname{tr}\varrho M)^{2}$. A quantum algorithm was proposed in [1] to test whether an unknown state is close to or far from a fixed state using $O(\varepsilon^{-2})$ copies of the state.

In order to test whether an unknown state is contained in or far from a given finite set of states, we need a revised tester. For the random variable $Y=\min\{X_{1},\dots,X_{m}\}$ where $X_{i}$’s are independent random variables, consider the tester which accepts (i.e., reports $\min_{i}\operatorname{\mathbb{E}}X_{i}\leq(1-2\gamma)\theta$) when $\min_{i}X_{i}\leq(1-\gamma)\theta$ and rejects (i.e., reports $\min_{i}\operatorname{\mathbb{E}}X_{i}\geq\theta$) otherwise.

When $\min\operatorname{\mathbb{E}}X_{i}\leq(1-2\gamma)\theta$, we have

where $\operatorname{\mathbb{E}}X_{k}=\min_{i}\operatorname{\mathbb{E}}X_{i}$.

When $\min\operatorname{\mathbb{E}}X_{i}\geq\theta$, we have

Using the above bounds, following the idea in [1], we have

We now apply the exponential Markov inequality to bound the deviation of $X$ from its expectation, yielding an alternative sample complexity for testing the property of a finite set of pure states. When $\operatorname{\mathbb{E}}X\leq(1-2\gamma)\theta$, the tester accepts with probability

for any $s>0$. When $\operatorname{\mathbb{E}}X\geq\theta$, then the tester rejects with probability

for any $s<0$.

Denote $\delta:=\{\textnormal{D}(\varphi_{1},\varphi_{2}):\varphi_{1}\neq\varphi_{2}\in\mathcal{P}\}$. It was shown in [20] that $O(\max\{\varepsilon^{-2},\delta^{-2}\}\ln|\mathcal{P}|)$ copies of states suffice to distinguish whether $\psi\in\mathcal{P}$ or $\textnormal{D}(\psi,\mathcal{P})\geq\varepsilon$. When $\delta=o(\varepsilon^{2})$, our method is more sample-efficient than that in [20]. The method based on Schur-Weyl decomposition used in this chapter may be useful in the certification in distributed scenarios.

Following [14], define the distance between two unitary matrices $U,V$ of order $d$ as

where $\langle{U,V}\rangle:=\operatorname{tr}(U^{\dagger}V)$. It can be seen that the distance of two unitaries is at most 1.

Notice that the normalized inner product of unitaries is equal to the inner product of their Choi states, i.e.,

Here and in the following we use $|{\phi^{\textnormal{m}}}\rangle$ to denote the (normalized) maximally entangled state. Consider the following Schur-Weyl decomposition

where $\mathcal{H}_{\lambda}$ is the irrep of $\textnormal{U}(d)$ and $\mathcal{K}_{\lambda}$ is its corresponding multiplicity space. The entanglement between the representation space and multiplicity space was used to estimate the group transformation [3]. It turns out that it can be also used in the certification of unitaries as shown in the following.

When both single-qubit unitaries are unknown, we have the following.

In order to deal with the higher dimensional case, an upper bound on $|\operatorname{tr}U_{\lambda}|/d_{\lambda}$ is needed given $|\operatorname{tr}U|/d\leq 1-\varepsilon^{2}$, where $U_{\lambda}$ is representation matrix of $U$ on irrep $\mathcal{H}_{\lambda}$ for appropriate $\lambda\in\textnormal{Par}(n)$ and $d_{\lambda}$ is the dimension of $\mathcal{H}_{\lambda}$.

For the asymptotic case where the value of $\varepsilon$ is small, $n$ will be so large that the irrep $\mathcal{H}_{\lambda}$ has smaller dimension than its multiplicity space $\mathcal{K}_{\lambda}$ does. Thus there exists a maximally entangled state $\phi^{\textnormal{m}}$ on $\mathcal{H}_{\lambda}\otimes\mathcal{H}_{\lambda}$ which is a subspace of $\mathcal{H}_{\lambda}\otimes\mathcal{K}_{\lambda}$. If $\varepsilon$ is not small enough and thus $n$ is not large enough, one can introduce a reference system of dimension $\frac{\dim\mathcal{H}_{\lambda}}{\dim\mathcal{K}_{\lambda}}$ to make them have equal dimension [3]. The dimension of the ancilla system, however, has been exponentially decreased compared with the usual method using Choi states directly. Since it is common to deal with low-dimensional quantum systems in quantum computing under current technology, our method exhibits advantage in practice.

We have proposed and analyzed in this work efficient algorithms for certification of quantum state set and quantum unitary.

For the certification of identity of qudit unitaries, we have considered the irrep corresponding partition $\lambda=(d-1,d-2,\dots,0)n/\binom{d}{2}$. One issue that would be worth investigating is whether this irrep is optimal compared with other irreps.

Using Theorem 6, one can also efficiently test whether one unitary is identity, and whether two unitaries are inverse to each other. It is known that many properties of quantum channel can be reduced to testing properties of quantum state via the Choi-Jamiołkowski isomorphism, but by employing this approach one usually needs to use quantum channels too many times and also to introduce extra ancilla system. We used group representation to test properties of unitaries more sample-efficiently without using ancilla system. The method based on group representation theory may be useful in testing other properties of quantum states and unitaries, which deserves further study. Our approach may be promising in the property testing of noisy quantum operation and measurement to achieve better performance, which remains further study.

This work was supported by the School of Computer Science and Technology, University of Science and Technology of China (Grant No. KY0110009999).