Hardness of classically sampling one clean qubit model with constant total variation distance error

TL;DR

This paper investigates the classical hardness of sampling from the one clean qubit (DQC1) model within a constant total variation distance error, showing it is possible under a modified average case hardness conjecture.

Contribution

It demonstrates that classical sampling hardness results for the DQC1 model can be extended to total variation distance error, given a modified hardness conjecture.

Findings

Sampling from DQC1 is classically hard under a modified conjecture.

Anti-concentration property holds due to small output probabilities.

Hardness results extend to total variation distance error.

Abstract

The one clean qubit model (or the DQC1 model) is a restricted model of quantum computing where only a single input qubit is pure and all other input qubits are maximally mixed. In spite of the severe restriction, the model can solve several problems (such as calculating Jones polynomials) whose classical efficient solutions are not known. Furthermore, it was shown that if the output probability distribution of the one clean qubit model can be classically efficiently sampled with a constant multiplicative error, then the polynomial hierarchy collapses to the second level. Is it possible to improve the multiplicative error hardness result to a constant total variation distance error one like other sub-universal quantum computing models such as the IQP model, the Boson Sampling model, and the Fourier Sampling model? In this paper, we show that it is indeed possible if we accept a modified version of the average case hardness conjecture. Interestingly, the anti-concentration lemma can be easily shown by using the special property of the one clean qubit model that each output probability is so small that no concentration occurs.

Equations

$$|p_z-q_z|\le ϵp_z$$

$$\sum _z|p_z-q_z|\le ϵ$$

$$f(z,U)\equiv ⟨z|U(|0⟩⟨0|\otimes I^{\otimes n})U^{†}|z⟩$$

$$U^{†}$$

$$f(0^{n+1},U)$$

$$⟨0^n|C|0^n⟩$$

$$U_1^{†}$$

$$U_2^{†}$$

$$f(0^{n+1},U_1)$$

$$f(0^{n+1},U_2)$$

$$|f(0^{n+1},U_1)-a|$$

$$|f(0^{n+1},U_2)-b|$$

$$P_{V_x}(o=0|p=0)=\frac{P_{V_x}(o=0,p=0)}{P_{V_x}(p=0)},$$

$$\frac{b}{a}$$

$$\sum _{z\in \{0,1{\}}^{n+1}}|p_z(U)-q_z(U)|\le ϵ$$

$$\displaystyle p_{z}(U)\equiv\langle z|U\Big{(}|0\rangle\langle 0|\otimes\frac{I^{\otimes n}}{2^{n}}\Big{)}U^{\dagger}|z\rangle.$$

$$\displaystyle\mbox{Pr}_{z,U}\Big{[}|p_{z}(U)-q_{z}(U)|\geq\frac{\epsilon}{2^{n+1}\delta}\Big{]}$$

$$|{\tilde{q}}_z(U)-q_z(U)|\le \frac{q_z(U)}{poly}.$$

$$|{\tilde{q}}_z(U)-p_z(U)|$$

$$\frac{ϵ}{2^{n+1}\delta }\le \frac{p_z(U)}{3}.$$

$$p_z(U)$$

$$\sum _{z\in \{0,1{\}}^{n+1}}{p}_z(U)=1$$

$$1$$

$$\frac{|S|}{2^{n+1}|{\mathcal{U}}_{n+1}|}$$

$$|{\tilde{q}}_z(U)-p_z(U)|$$

$$F\equiv 1-\delta -\frac{1}{2-\frac{3ϵ}{\delta }}$$

$$F=\frac{1}{6}.$$

$$p_z(U)=\frac{f(z,U)}{2^n}.$$

$$|{\tilde{q}}_z(U)2^n-f(z,U)|$$

Full text

I Introduction

The one clean qubit model (or the DQC1 model) first introduced by Knill and Laflamme KnillLaflamme is a restricted model of quantum computing where only a single input qubit is pure and all other input qubits are maximally mixed. In spite of the severe restriction, surprisingly, the model can solve several problems whose efficient classical solutions are not known, such as the spectral density estimation KnillLaflamme , testing integrability integrability , calculations of the fidelity decay fidelity_decay , and approximations of the Jones polynomial, HOMFLY polynomial, and Turaev-Viro invariant ShorJordan08 ; Passante09 ; JordanWocjan09 ; JordanAlagic . Furthermore, it was recently shown that if the output probability distribution of the one clean qubit model is classically efficiently sampled with a constant multiplicative error, then the polynomial hierarchy collapses to the second level MFF ; FKMNTT . (Here, we say that a probability distribution $\{p_{z}\}_{z}$ is sampled by a machine $M$ with a multiplicative error $\epsilon\geq 0$ if

[TABLE]

is satisfied for all $z$, where $\{q_{z}\}_{z}$ is the output probability distribution of $M$.) Since a collapse of the polynomial hierarchy is not believed to happen in computer science, the result suggests the impossibility of classically simulating the one clean qubit model. Similar hardness results for constant multiplicative error sampling were also shown for other sub-universal quantum computing models, such as the IQP model BJS and the Boson Sampling model AA .

The requirement of constant multiplicative error sampling is, however, strong, and sampling with a constant total variation distance error (or the L1-norm error) is considered as more appropriate. (Here, we say that a probability distribution $\{p_{z}\}_{z}$ is sampled by a machine $M$ with a total variation distance error $\epsilon\geq 0$ if

[TABLE]

is satisfied, where $\{q_{z}\}_{z}$ is the output probability distribution of $M$.) In fact, the hardness results with constant total variation distance errors were shown for the IQP model BMS , the Boson Sampling model AA , and the Fourier Sampling model FU (assuming some conjectures). Is it possible to show a similar constant-total-variation-distance-error hardness result for the one clean qubit model?

In this paper, we show that it is indeed possible if we accept a modified version of the average case hardness conjecture. Our proof is similar to those of Refs. AA ; BMS ; FU , but there is one interesting difference which is specific to the one clean qubit model: the anti-concentration lemma can be easily shown. For the Boson Sampling model and the Fourier Sampling model, the anti-concentration lemma is a conjecture AA ; FU . For the IQP model, it is shown with some calculations by using a special structure of IQP circuits BMS . For the present case, as we will see later, the anti-concentration lemma is easily shown by using a special property of the one clean qubit model that each probability is so small that no concentration occurs.

II Average case hardness conjecture

As in the cases of other sub-universal quantum computing models, such as the IQP model BMS , the Boson Sampling model AA , and the Fourier Sampling model FU , we need a conjecture so-called “average case hardness conjecture”, which claims that the $\#$P-hardness for the worst case can be lifted to an average case. To show our result, which is a hardness of efficient classical sampling of the one clean qubit model with a constant total variation distance error, we need the following conjecture.

Conjecture: For each $n$, there exists a discrete set ${\mathcal{U}}_{n+1}$ of uniformly-generated polynomial-time $(n+1)$-qubit unitary operators such that calculating

[TABLE]

with a multiplicative error less than 1/2 for more than 1/6 fraction of $(z,U)\in\{0,1\}^{n+1}\times{\mathcal{U}}_{n+1}$ is $\#$P-hard.

Here, $I\equiv|0\rangle\langle 0|+|1\rangle\langle 1|$ is the two-dimensional identity operator, and $|z\rangle$ is the computational-basis state corresponding to the bit string $z$. (For example, if $z=010$, $|z\rangle=|010\rangle=|0\rangle\otimes|1\rangle\otimes|0\rangle$.)

Unfortunately, we do not know whether the conjecture is true or not, but we can show that it is true at least for the worst case. Here, we give two proofs for the worst-case $\#$P-hardness.

II.1 First proof

Let us consider a unitary operator $U$ such that

[TABLE]

where $C$ is an $n$-qubit IQP circuit. Then,

[TABLE]

For certain IQP circuits $C$, $\langle 0^{n}|C|0^{n}\rangle$ is related to the partition function, $Z$, of the Ising model BMS ; FujiiMorimae and the gap function, $gap(f)$, of a degree-3 polynomial $f$ over ${\mathbb{F}}_{2}$ BMS :

[TABLE]

It is known that calculating $|Z|^{2}$ and $gap(f)^{2}$ with constant multiplicative errors is $\#$P-hard BMS ; FujiiMorimae . Hence calculating $|\langle 0^{n}|C|0^{n}\rangle|^{2}=f(0^{n+1},U)$ with constant multiplicative errors is $\#$P-hard for certain unitary operators $U$, which shows the correctness of the conjecture for the worst case.

II.2 Second proof

We define two unitary operators $U_{1}$ and $U_{2}$ as

[TABLE]

where $V$ is an $n$-qubit unitary operator. Then, we obtain

[TABLE]

Now we show that calculating $f(0^{n+1},U_{1})$ and $f(0^{n+1},U_{2})$ with a constant multiplicative error $0\leq\epsilon<1$ is postBQP-hard. Since ${\rm postBQP}={\rm PP}$ postBQP and ${\rm P}^{\rm PP}={\rm P}^{\#{\rm P}}$, it means that the calculation is $\#$P-hard. Proof is as follows. Let us assume that there exists an algorithm that calculates $a$ and $b$ such that

[TABLE]

Let $L$ be a language in postBQP. Then, for any polynomial $r$, there exists a uniform family $\{V_{x}\}_{x}$ of polynomial-time quantum circuits such that

• •

If $x\in L$ then $P_{V_{x}}(o=0|p=0)\geq 1-2^{-r}$.

• •

If $x\notin L$ then $P_{V_{x}}(o=0|p=0)\leq 2^{-r}$.

Here,

[TABLE]

$P_{V_{x}}(o=0,p=0)$ is the probability that $V_{x}$ outputs $(o,p)=(0,0)$, and $P_{V_{x}}(p=0)$ is the probability that $V_{x}$ outputs $p=0$.

Let us construct $U_{1}$ and $U_{2}$ by using $V_{x}$. Then, for any polynomial $r$, if $x\in L$,

[TABLE]

and if $x\notin L$,

[TABLE]

Therefore, if we can calculate $a$ and $b$, we can solve $L$.

III Main result

If we accept the conjecture, we can show the following theorem, which is the main result of the present paper.

Theorem: If there exists a probabilistic polynomial-time classical algorithm that outputs $z\in\{0,1\}^{n+1}$ with probability $q_{z}(U)$ such that

[TABLE]

for any $U\in\cup_{n}{\mathcal{U}}_{n+1}$, then the polynomial hierarchy collapses to the third level. Here, $\epsilon=\frac{1}{36}$ and

[TABLE]

The theorem says that if the output probability distribution $p_{z}(U)$ of the one clean qubit model can be classically efficiently sampled with the total variation distance error $\epsilon$, then the polynomial hierarchy collapses to the third level.

Proof: Now let us give a proof of the theorem. Our proof is similar to those of Refs. AA ; BMS ; FU except that the anti-concentration lemma can be easily shown.

Let $\delta>0$ be a parameter specified later. From Markov’s inequality,

[TABLE]

From Stockmeyer’s Counting Theorem Stockmeyer , there exists an ${\rm FBPP}^{\rm NP}$ algorithm that outputs $\tilde{q}_{z}(U)$ such that

[TABLE]

Therefore,

[TABLE]

with more than $1-\delta$ fraction of $(z,U)$.

Let $S\subseteq\{0,1\}^{n+1}\times{\mathcal{U}}_{n+1}$ be the set of $(z,U)$ such that

[TABLE]

Since

[TABLE]

for all $(z,U)$, and

[TABLE]

for all $U$, we obtain

[TABLE]

which means

[TABLE]

Therefore,

[TABLE]

for more than

[TABLE]

fraction of $(z,U)$. For example, if we take $\delta=6\epsilon$,

[TABLE]

Note that

[TABLE]

Therefore, the above result means that there exists an ${\rm FBPP}^{\rm NP}$ algorithm that outputs $\tilde{q}_{z}(U)$ such that

[TABLE]

for more than 1/6 fraction of $(z,U)$. If our average-case hardness conjecture is true, it means the collapse of the polynomial hierarchy to the third level.

Acknowledgements.

TM is supported by JST ACT-I No.JPMJPR16UP, the Grant-in-Aid for Scientific Research on Innovative Areas No.15H00850 of MEXT Japan, and the JSPS Grant-in-Aid for Young Scientists (B) No.26730003 and No.17K12637.