Verified measurement-based quantum computing with hypergraph states

TL;DR

This paper introduces a verification method for hypergraph states using only sequential single-qubit Pauli measurements, enabling secure blind quantum computing and quantum supremacy demonstrations.

Contribution

It presents the first verification protocol for hypergraph states, expanding measurement-based quantum computing capabilities.

Findings

Hypergraph states can be verified with simple measurements.

Hypergraph states enable universal measurement-based quantum computing.

Verification supports secure blind quantum computing and quantum supremacy.

Abstract

Hypergraph states are generalizations of graph states where controlled-$Z$ gates on edges are replaced with generalized controlled-$Z$ gates on hyperedges. Hypergraph states have several advantages over graph states. For example, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements, while graph state measurement-based quantum computing needs non-Clifford basis measurements. Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph states with a constant $L$1-norm error unless the polynomial hierarchy collapses to the third level. Although several protocols have been proposed to verify graph states with only sequential single-qubit Pauli measurements, there was no verification method for hypergraph states. In this paper, we propose a method for verifying hypergraph states with only sequential single-qubit Pauli measurements. As applications, we consider verified blind quantum computing with hypergraph states, and quantum supremacy demonstrations with hypergraph states.

Equations

$$\displaystyle|G\rangle\equiv\Big{(}\prod_{e\in E}\widetilde{CZ}_{e}\Big{)}|+\rangle^{\otimes n}$$

$$g_i$$

$$W_i^Z$$

$$W_i^{CZ}$$

$$g_i$$

$$s_t$$

$$\displaystyle s_{t}=(-1)^{\alpha_{t}}X_{i}\Big{(}\prod_{j\in D_{t}}Z_{j}\Big{)}.$$

$$\displaystyle p_{{\rm test},i}\equiv\frac{1}{4^{r}}\sum_{t\in\{1,2,3,4\}^{r}}\mbox{Tr}\Big{(}\rho\frac{I^{\otimes n}+s_{t}}{2}\Big{)}=\frac{1}{2}+\frac{\mbox{Tr}(\rho g_{i})}{2^{r+1}}.$$

$$\text{mbox}Pr[\text{mbox}AliceacceptsBob]$$

$$\displaystyle\mbox{Tr}\Big{[}(T\otimes\Pi_{G}^{\perp})\rho^{\otimes nk+1}\Big{]}\leq\frac{1}{2n^{2}}.$$

$$\displaystyle\mbox{Tr}\Big{[}(T\otimes\Pi_{G}^{\perp})\Psi^{\prime}\Big{]}$$

$$1-⟨G|\rho |G⟩$$

$$\displaystyle\mbox{Tr}\Big{[}(T\otimes\Pi_{G}^{\perp})\rho^{\otimes nk+1}\Big{]}$$

$$p_{\mathrm{test},i}=\frac{1}{2}+\frac{\text{mbox}Tr(\rho g_i)}{2^{r+1}}<\frac{1}{2}+\frac{1-\delta }{2^{r+1}}$$

$$\text{mbox}Tr[(T\otimes I){\rho }^{\otimes nk+1}]$$

$$\text{mbox}Tr[(T\otimes {\Pi }_G^{\perp }){\rho }^{\otimes nk+1}]$$

$$\displaystyle\mbox{Tr}[(T\otimes\Pi_{G}^{\perp})\rho^{\otimes nk+1}]\leq\max\Big{(}\frac{1}{2n^{2}},e^{-n}\Big{)}=\frac{1}{2n^{2}}.$$

$$\sum _z|p_z-q_z|$$

Full text

Many-point correlations in quantum many-body systems are one of the most essential ingredients in condensed-matter physics and statistical physics. Correlations of sequential single-qubit measurements on quantum states are also important drive forces for quantum information processing. For example, measurement-based quantum computing MBQC , which is nowadays one of the standard quantum computing models, enables universal quantum computing with only adaptive single-qubit measurements on certain quantum states, such as graph states MBQC and other condensed-matter-physically motivated states including the AKLT state BrennenMiyake ; MBQC1 ; MBQC2 ; MBQC3 ; MBQC4 ; MBQC5 ; MBQC6 ; MBQC7 ; MBQC8 ; MBQC9 ; MBQC10 ; MBQC11 ; MBQC12 ; MBQC13 ; MBQC14 ; MBQC15 . Furthermore, not only adaptive but also non-adaptive single-qubit measurements on graph states can demonstrate a quantumness which cannot be classically efficiently simulated: it is known that if probability distributions of non-adaptive sequential single-qubit measurements on graph states are classically efficiently sampled, then the polynomial hierarchy collapses to the third level IQP1 ; IQP2 ; FujiiMorimae or the second level KobayashiICALP . The polynomial hierarchy is a hierarchy of complexity classes generalizing P and NP, and it is not believed to collapse in computer science. It is an example of recently well studied “quantum supremacies” of sub-universal quantum computing models, which are expected to be easier to experimentally implement, but can outperform classical computing. (For details, see Refs. bosonsampling ; bosonsampling2 ; MFF ; KobayashiICALP ; FujiiMorimae ; IQP1 ; IQP2 and their supplementary materials.)

For practical implementations of measurement-based quantum computing and experimental demonstrations of the quantum supremacy, verifying graph states is essential, since in reality a generated state cannot be the ideal graph state due to some experimental noises. The problem becomes more serious if we consider delegated secure quantum computing, so called blind quantum computing BFK ; FK . It is known that the ability of sequentially measuring single qubits is enough to secretly delegate quantum computing to a remote server measuringAlice ; HayashiMorimae . The honest server sends each qubit of a graph state one by one to the user, and user can realize any quantum computing with only sequential single-qubit measurements. If the server is malicious, however, a completely wrong state might be sent to the user. The user therefore needs to test the state sent from the server. In such a quantum cryptographic scenario, the situation is worse than the single-party laboratory experiments, since the noises on the given state are caused by malicious servers and therefore not necessarily physically natural ones. Several methods of verifying graph states with only sequential single-qubit Pauli measurements have been proposed HayashiMorimae ; MorimaeNagajSchuch . (If more than two non-communicating servers are available, a completely classical user can verify stabilizer states VaziraniNature ; Matt ; Ji .) In the protocol of Ref. HayashiMorimae , the user does a test so called the stabilizer test on some parts of the state sent from the server. The stabilizer test can be done with only sequential single-qubit Pauli measurements. If the user passes the test, the remaining state is guaranteed to be close to the ideal graph state.

Since the protocol of Ref. HayashiMorimae makes no assumption (such as the i.i.d. sample or physically natural noises) on the given state, the verification method can be used in quantum cryptographic contexts. In particular, verified blind quantum computing and verified quantum supremacy demonstrations can be realized with graph states verified through the protocol. There are, however, two problems. First, in the verified blind protocol of Ref. HayashiMorimae , the user needs non-Clifford basis measurements for computing (the verification itself can be done with only Pauli measurements). It would be better if both the verification and the computation can be done with only Pauli measurements LH . Second, the quantum supremacy demonstration with graph states IQP1 , which needs only non-adaptive measurements, requires somehow a strict approximation, namely a multiplicative-error approximation.

Recently, two breakthroughs that solve these drawbacks of graph states have been done. These results use hypergraph states Rossi ; HG1 ; HG2 ; HG3 ; HG4 in stead of graph states. (For the definition of hypergraph states and their properties, see below.) First, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements MillerMiyake . This result solves the first problem, namely, the requirement of non-Clifford basis measurements for the user. Therefore, by using the hypergraph states, the one-way secure delegated quantum computing is possible for the user who can do only Pauli measurements. Ref. MillerMiyake also pointed out that hypergraph states are important in the study of symmetry-protected topological orders. Second, it was shown in Ref. IQP2 that if hypergraph states are considered, the multiplicative error requirement can be replaced with an $L$1-norm one, which is more relaxed. This result solves the second problem.

In short, hypergraph states are promising novel resource states for many quantum information processing tasks. Unfortunately, however, there was no verification method for hypergraph states. In particular, we did not know how to test a given hypergraph state with only sequential single-qubit Pauli measurements. It was a huge obstacle for practical applications of hypergraph states in quantum information and condensed matter physics.

In this paper, we propose a method for verifying hypergraph states with only sequential single-qubit Pauli measurements. As in the case of the graph state verification HayashiMorimae , the user does a certain test on some parts of the state sent from the server. If the user passes the test, then the remaining state is guaranteed to be close to the ideal hypergraph state. As applications, we consider verified blind quantum computing with hypergraph states, and verified quantum supremacy demonstrations with hypergraph states.

Hypergraph states.— We first define hypergraph states, and explain their properties. A hypergraph $G\equiv(V,E)$ is a pair of a set $V$ of vertices and a set $E$ of hyperedges, where $n\equiv|V|$. A hyperedge may link more than two vertices. For simplicity, in this paper, we assume that $2\leq|e|\leq 3$ for all $e\in E$, where $|e|$ is the number of vertices linked to the hyperedge $e$. (Generalizations to other cases would be possible.) Let

[TABLE]

be the hypergraph state corresponding to the hypergraph $G$, where $\widetilde{CZ}_{e}\equiv\bigotimes_{i\in e}I_{i}-2\bigotimes_{i\in e}|1\rangle\langle 1|_{i}$ is the generalized $CZ$ gate acting on vertices in the hyperedge $e$. Here, $I$ is the two-dimensional identity operator. For example, if $|e|=2$, it is nothing but the standard $CZ$ gate. If $|e|=3$, it is the $CCZ$ gate, $CCZ\equiv(I^{\otimes 2}-|11\rangle\langle 11|)\otimes I+|11\rangle\langle 11|\otimes Z.$

The stabilizer $g_{i}$ of $|G\rangle$ associated with the vertex $i$ is defined by

[TABLE]

where

[TABLE]

It is easy to check that the following properties are satisfied: $[g_{i},g_{j}]=0$ for all $i,j\in V$. $g_{i}|G\rangle=|G\rangle$ for all $i\in V$. $g_{i}^{2}=I^{\otimes n}$ for all $i\in V$. $\prod_{i=1}^{n}\frac{I^{\otimes n}+g_{i}}{2}=|G\rangle\langle G|.$

Stabilizer test for $g_{i}$.— Before introducing our verification protocol, we define the stabilizer test for each $g_{i}$, which is an essential ingredient of the protocol. Note that $CZ_{j,k}=\frac{1}{2}(I_{j}\otimes I_{k}+I_{j}\otimes Z_{k}+Z_{j}\otimes I_{k}-Z_{j}\otimes Z_{k}).$ Therefore

[TABLE]

where $r\equiv|W_{i}^{CZ}|$, $t\equiv\{t_{j,k}\}_{(j,k)\in W_{i}^{CZ}}$, $\sigma_{j,k}(1)\equiv I_{j}\otimes I_{k}$, $\sigma_{j,k}(2)\equiv I_{j}\otimes Z_{k}$, $\sigma_{j,k}(3)\equiv Z_{j}\otimes I_{k}$, $\sigma_{j,k}(4)\equiv-Z_{j}\otimes Z_{k}$, and

[TABLE]

Let us define a bit $\alpha_{t}\in\{0,1\}$ and a subset $D_{t}\subseteq V$ such that

[TABLE]

Note that $\alpha_{t}$ and $D_{t}$ can be calculated in polynomial time. In fact, $\alpha_{t}$ can be calculated in the following way. We first set $\alpha_{t}=0$. If $t_{j,k}=4$, we flip $\alpha_{t}$. We do it for all $(j,k)\in W_{i}^{CZ}$. Since $|W_{i}^{CZ}|\leq{n-1\choose 2}=O(n^{2})$, it takes at most polynomial time. Furthermore, $D_{t}$ can be calculated in the following way. We first set $D_{t}=W_{i}^{Z}$. We then update $D_{t}$ according to $t_{j,k}$ for each $(j,k)\in W_{i}^{CZ}$. Again, $|W_{i}^{CZ}|\leq O(n^{2})$ means that it takes at most polynomial time.

Let $\rho$ be an $n$-qubit state. We define the “stabilizer test for $g_{i}$ on $\rho$” as the following Alice’s action:

Alice randomly generates $t\in\{1,2,3,4\}^{r}$.

• 2.

She measures $i$th vertex of $\rho$ in $X$, and $j$th vertex of $\rho$ in $Z$ for all $j\in D_{t}$.

Let $x\in\{+1,-1\}$ be the measurement result of the $X$ measurement, and $z_{j}\in\{+1,-1\}$ be that of the $Z$ measurement on vertex $j\in D_{t}$. We say that Alice passes the stabilizer test for $g_{i}$ on $\rho$ if $x\prod_{j\in D_{t}}z_{j}=(-1)^{\alpha_{t}}.$

The probability $p_{{\rm test},i}$ that Alice passes the stabilizer test for $g_{i}$ on $\rho$ is expr

[TABLE]

Verification protocol.— We now explain our verification protocol. Bob sends Alice an $n(nk+1+m)$-qubit state $\Psi$, where $k=2^{2r+3}n^{7}$ and $m\geq 2n^{7}k^{2}\ln 2$. The state $\Psi$ consists of $nk+1+m$ registers (Fig. 1). Each register stores $n$ qubits. (If Bob is honest, every register is in the state $|G\rangle$. If Bob is malicious, on the other hand, $\Psi$ can be any $n(nk+1+m)$-qubit entangled state.) Alice randomly permutes registers and discards $m$ registers. (As we will see later, this random permutation and discarding of some registers are necessary to guarantee that the remaining state is close to an i.i.d. sample by using the quantum de Finetti theorem LiSmith .) Let $\Psi^{\prime}$ be the remaining state. The state $\Psi^{\prime}$ consists of $nk+1$ registers. She chooses one register from $\Psi^{\prime}$, which is used for the measurement-based quantum computing. We call the register computing register. The remaining $nk$ registers of $\Psi^{\prime}$ are divided into $n$ groups. Each group consists of $k$ registers. The stabilizer test for $g_{i}$ is performed on every register in the $i$th group for $i=1,2,...,n$. (Note that Alice does not need to do the permutation “physically”, which requires a quantum memory. Bob just sends each qubit of $\Psi$ one by one to Alice, and Alice randomly chooses her action from the test, discarding, or computation.)

Let $K_{i}$ be the number of times that Alice passes the stabilizer test for $g_{i}$, i.e. the random variable to describe the number of Alice’s observation of the event $\frac{1}{4^{r}}\sum_{t}\frac{I^{\otimes n}+s_{t}}{2}$. If $\frac{K_{i}}{k}\geq\frac{1}{2}+\frac{1-\epsilon}{2^{r+1}},$ we say that the $i$th group passes the test. Here, $\epsilon=\frac{1}{2n^{3}}$. If all groups pass the test, we say that Alice accepts Bob.

The main results of the present paper are the following two items:

1. Completeness: if every register of $\Psi$ is in the state $|G\rangle$, then the probability that Alice accepts Bob is larger than $1-ne^{-n}$.

2. Soundness: if Alice accepts Bob, the state $\rho_{\rm comp}$ of the computing register satisfies $\langle G|\rho_{\rm comp}|G\rangle\geq 1-\frac{1}{n}$ with a probability larger than $1-\frac{1}{n}$.

Proof of the completeness.— We first show the completeness. If every register of $\Psi$ is in the state $|G\rangle$, then $p_{{\rm test},i}=\frac{1}{2}+\frac{1}{2^{r+1}}$ for all $i=1,2,...,n$. From the union bound and the Hoeffding inequality,

[TABLE]

Proof of the soundness.— We next show the soundness. We define the $n$-qubit projection operator $\Pi_{G}^{\perp}\equiv I^{\otimes n}-|G\rangle\langle G|.$ Let $T$ be the POVM element corresponding to the event that Alice accepts Bob. We can show that for any $n$-qubit state $\rho$,

[TABLE]

Its proof is given later. Due to the quantum de Finetti theorem (for the one-way LOCC norm version) LiSmith ,

[TABLE]

We have $\mbox{Tr}[(T\otimes\Pi_{G}^{\perp})\Psi^{\prime}]=\mbox{Tr}(\Pi_{G}^{\perp}\rho_{\rm comp})\mbox{Tr}[(T\otimes I)\Psi^{\prime}].$ Therefore, if $\mbox{Tr}(\Pi_{G}^{\perp}\rho_{\rm comp})>\frac{1}{n}$, then $\mbox{Tr}[(T\otimes I)\Psi^{\prime}]<\frac{1}{n},$ which means that if Alice accepts Bob, $\langle G|\rho_{\rm comp}|G\rangle\geq 1-\frac{1}{n}$ with a probability larger than $1-\frac{1}{n}$.

Proof of Eq. (1).— First, let us assume that $\mbox{Tr}(\rho g_{i})\geq 1-\delta$ for all $i=1,2,..,n$, where $\delta=\frac{1}{n^{3}}$. Due to the union bound,

[TABLE]

Therefore,

[TABLE]

Next let us assume that $\mbox{Tr}(\rho g_{i})<1-\delta$ for at least one $i$. In this case,

[TABLE]

for the $i$. Then, due to the Hoeffding inequality,

[TABLE]

Hence

[TABLE]

From Eqs. (2) and (3), for any state $\rho$,

[TABLE]

Applications.— To conclude this paper, we finally discuss two applications of our results. First, our verification protocol can be used in verified blind quantum computing. In the protocol of Ref. HayashiMorimae , the user needs non-Clifford basis measurements to implement quantum computing (the verification itself can be done with only Pauli measurements.) On the other hand, if the server generates the Union Jack states MillerMiyake , for example, the user needs only Pauli measurements for both the verification and the computation.

Second, our protocol can be used for the verified quantum supremacy demonstration. It was shown in Ref. IQP2 that the following is true for several hypergraph states (assuming the so called “worst case vs average case” conjecture): if there exists a classical sampler that outputs $z$ with probability $q_{z}$ such that $\sum_{z\in\{0,1\}^{n}}|p_{z}-q_{z}|\leq\frac{1}{192},$ then the polynomial hierarchy collapses to the third level. Here, $p_{z}$ is the probability of obtaining the result $z\in\{0,1\}^{n}$ when certain single-qubit measurements are done on an $n$-qubit hypergraph state. This result means that if we can generate hypergraph states, we can demonstrate the quantum supremacy. However, what happens if we cannot have the ideal hypergraph state, and only the verified state $\rho_{\rm comp}$ is available? (For example, Alice, who can do only single-qubit measurements, might want untrusted Bob to send a hypergraph state.) We can show that the state $\rho_{\rm comp}$ is enough to demonstrate the same quantum supremacy. In fact, let us assume that there exists a classical sampler such that $\sum_{z}|p_{z}^{\prime}-q_{z}|\leq\frac{1}{192},$ where $p^{\prime}_{z}$ is the output probability distribution of the single-qubit measurements on $\rho_{\rm comp}$. Then,

[TABLE]

which means that the classical sampler can also sample $p_{z}$ with the $\sim 1/192$ $L$1-norm error. For example, the hypergraph states naturally induced from the IQP circuits corresponding to the non-adaptive Union Jack state measurement-based quantum computing MillerMiyake can be used for that purpose. Since the non-adaptive Union Jack state measurement-based quantum computing is universal with postselections, a multiplicative error calculation of its output probability distribution is $\#$P-hard FujiiMorimae . If we assume the worst case hardness can be lifted to the average case one, we can show the hardness of the classical constant $L$1-norm error sampling.

TM is supported by the JST ACT-I, the JSPS Grant-in-Aid for Young Scientists (B) No.26730003, and the MEXT JSPS Grant-in-Aid for Scientific Research on Innovative Areas No.15H00850. YT is supported by the Program for Leading Graduate Schools: Interactive Materials Science Cadet Program. MH is supported in part by Fund for the Promotion of Joint International Research (Fostering Joint International Research) No.15KK0007, the JSPS MEXT Grant-in-Aid for Scientific Research (B) No.16KT0017, the Okawa Research Grant and Kayamori Foundation of Information Science Advancement.