I Introduction
Quantum computing is believed to outperform classical computing. In fact, quantum advantages have been shown in terms of, for example, the communication complexity comm1 ; comm2 . Regarding the time complexity, however, the ultimate goal, , seems to be extremely hard to show because of the barrier complexity_zoo .
Mainly three types of approaches exist to demonstrate quantum speedups over classical computing. First approach is to construct quantum algorithms that are faster than known best classical algorithms. For example, quantum computing can do factoring Shor and simulations of quantum manybody dynamics Iulia , etc. faster than known best classical algorithms. (Classical best algorithms can be, however, updated Tang .) Second approach is to study the query complexity. Quantum computing has been shown to require fewer oracle queries than classical computing Simon ; Grover .
Third one, which has been actively studied recently, is to reduce classical simulatabilities of quantum computing (in terms of sampling) to certain unlikely collapses of conjectures in classical computational complexity theory, such as the infiniteness of the polynomialtime hierarchy. Several subuniversal quantum computing models have been proposed, such as the depthfour circuits TD , the Boson Sampling model BS , the IQP model IQP1 ; IQP2 , the oneclean qubit model (or the DQC1 model) KL ; MFF ; M ; Kobayashi ; KobayashiICALP , the random gate model random , and the HC1Q model HC1Q . (Definitions of the IQP model, DQC1 model, and HC1Q model are given later.) It has been shown that if output probability distributions of these models are classically sampled in polynomialtime, then the polynomialtime hierarchy collapses. The polynomialtime hierarchy is not believed to collapse in classical complexity theory, and therefore if we conjecture the infiniteness of the polynomialtime hierarchy, the reductions suggest quantum computational supremacy of those subuniversal models.
In this paper, we first focus on the DQC1 model, which is defined as follows.
Definition 1
The qubit DQC1 model with a unitary is the following quantum computing model.

The initial state is
where is the twodimensional identity operator.

The unitary operator is applied on the initial state to generate the state

The first qubit is measured in the computational basis. If the output is 0 (1), then accept (reject).
Note that throughout this paper we consider only size without explicitly mentioning it. The DQC1 model was originally introduced by Knill and Laflamme to model the NMR quantum computing KL , and since then many results have been obtained on the model KL ; Poulin1 ; Poulin2 ; ShorJordan ; Passante ; JordanWocjan ; JordanAlagic ; MFF ; M ; Kobayashi ; nonclean ; KobayashiICALP
. For example, the DQC1 model can solve several problems whose classical efficient solutions are not known, such as the spectral density estimation
KL , testing integrability Poulin1 , calculations of the fidelity decay Poulin2 , and approximations of Jones and HOMFLY polynomials ShorJordan ; Passante ; JordanWocjan . The acceptance probability of the qubit DQC1 model with a unitary isIt is known that if is classically sampled in polynomialtime within a multiplicative error , then the polynomialtime hierarchy collapses to the second level Kobayashi ; KobayashiICALP .
Definition 2
We say that is classically sampled in time within a multiplicative error if there exists a classical probabilistic algorithm that runs in time such that
where is the acceptance probability of the classical algorithm.
In this way, previous quantum supremacy results prohibit classical polynomialtime sampling of the DQC1 model based on the infiniteness of the polynomialtime hierarchy. A shortcoming is, however, that a possibility of exponentialtime or even superpolynomialtime classical sampling is not excluded: the acceptance probability of the qubit DQC1 model could be classically sampled in, say, time.
In this paper, we show finegrained quantum supremacy of the DQC1 model. We assume certain complexity conjectures, and show that the acceptance probability of the DQC1 model cannot be classically sampled within a multiplicative error in certain exponential or subexponential time (depending on conjectures). More precisely, in Sec. II, we introduce three conjectures, Conjecture 1, Conjecture 2, and Conjecture 3. Conjecture 1 seems to be more stable than Conjecture 2, and Conjecture 2 seems to be more stable than Conjecture 3. Conjecture 1 prohibits classical sampling of the qubit DQC1 model in subexponential time of (Theorem 1), while other two conjectures prohibit classical sampling of the qubit DQC1 model in time for any (Theorem 3 and Theorem 4).
We also show similar finegrained quantum supremacy results for another subuniversal quantum computing model, namely, the Hadamardclassical circuit with onequbit (HC1Q) model, which is defined as follows.
Definition 3
The qubit HC1Q model with a classical reversible circuit is the following quantum computing model.

The initial state is , where means .

The operation
is applied on the initial state. Here,
is the Hadamard gate, and the classical circuit is applied “coherently”.

Some qubits are measured in the computational basis.
Note that throughout this paper we consider only size without explicitly mentioning it. The HC1Q model is in the second level of the Fourier hierarchy FH2 where several useful circuits, such as those for Shor’s factoring algorithm Shor and Simon’s algorithm Simon , are placed. Furthermore, it was shown in Ref. HC1Q that output probability distributions of the HC1Q model cannot be classically sampled in polynomialtime within a multiplicative error unless the polynomialtime hierarchy collapses to the second level. We show that under the finegrained complexity conjectures, the output probability distributions of the HC1Q model cannot be classically sampled within a multiplicative error in certain exponential or subexponential time (depending on conjectures). As in the case of the DQC1 model, Conjecture 1 prohibits classical sampling of the qubit HC1Q model in subexponential time of (Theorem 2), while Conjecture 3 prohibits classical sampling of the qubit HC1Q model in time for any (Theorem 5).
There are some previous results AaronsonChen ; Huang ; Dalzell that showed quantum computational supremacy based on other conjectures than the infiniteness of the polynomialtime hierarchy. In particular, Ref. Dalzell showed finegrained quantum supremacy of the IQP model, QAOA model, and Boson Sampling model. For details, see Sec. IV.
Finally, in this paper, we also study universal quantum computing with Clifford and gates, where
and Clifford gates are ,
For an qubit quantum circuit that consists of only Clifford and gates, we define the acceptance probability by
Due to the GottesmanKnill theorem GK , can be exactly calculated in time if contains at most number of gates. The bruteforce classical calculation of needs time that scales in an exponential of , where is the number of gates.
In Sec. II, we introduce two conjectures, Conjecture 4 and Conjecture 5. Conjecture 4 is equivalent to socalled the exponentialtime hypothesis (ETH) IPZ01 . Under Conjecture 4, we show that cannot be classically calculated in time within an additive error smaller than (Theorem 6). Conjecture 5 is a version of Conjecture 4, and it allows us to show that cannot be classically sampled in time within a multiplicative error (Theorem 7). There is a classical algorithm that computes within a constant multiplicative error and in time with a nontrivial factor BravyiGosset . Our second result (Theorem 7) therefore suggests that improving to some subexponential time, say , is impossible. Note that Ref. DalzellPhD showed that must be under the conjecture,
where MAJZEROS is the problem of deciding whether or not for given a degree3 polynomial in variables over the field .
Theorem 6 considers the hardness of calculating output probability distributions (i.e., the strong simulation) in terms of scaling, while Ref. Huang considered that in terms of scaling. It was shown in Ref. Huang that if SETH is true then calculating output probability amplitudes of a certain qubit quantum computation within the additive error needs time. Note that the strong simulation, i.e., calculating output probability distributions of quantum computing within an exponentially small additive error, is Phard, which means that hard even for quantum computing. However, it is still meaningful to study strong simulations of quantum computing because several quantum computing models, such as Clifford circuits and matchgate circuits match , are known to be strongly simulatable.
The Paulibased computation (PBC) BravyiSmithSmolin is a universal quantum computing model closely related to the Clifford quantum computing, which is defined as follows.
Definition 4
The qubit Paulibased quantum computation (PBC) is the following quantum computing model.

The initial state is , where
is a magic state.

Nondestructive Pauli measurements are done adaptively.

The measurement results are finally classically processed.
Ii Conjectures
In this section, we introduce conjectures. First let us consider the following three conjectures.
Conjecture 1
Let be any nondeterministic time algorithm such that the following holds: given (a description of) a polynomialsize Boolean circuit , accepts if and rejects if , where
Then, for any constant , holds for infinitely many .
Conjecture 2
It is the same as Conjecture 1 except that the Boolean circuit is of logarithmic depth.
Conjecture 3
Let be any nondeterministic time algorithm such that the following holds: given (a description of) a polynomialsize classical reversible circuit that consists of only NOT and TOFFOLI, accepts if and rejects if , where
and is the last bit of . Then, for any constant , holds for infinitely many .
These conjectures are considered as strong (finegrained) versions of the conjecture, , which is believed because leads to the collapse of the polynomialtime hierarchy to the second level Kobayashi ; KobayashiICALP . Here, is defined as follows.
Definition 5
A language is in
if and only if there exists a nondeterministic polynomialtime Turing machine such that if
then the number of accepting paths is not equal to that of rejecting paths, and if then they are equal.These conjectures should be justified by the following arguments. First of all, it is true that direct connections between our conjectures and SETH IP01 ; IPZ01 (or NSETH CGIMPS ) are not clear, because acceptance criteria are different. (Our conjectures are based on gap functions, while SETH and NSETH are on
P functions.) However, at this moment, the only known way of deciding
or is to solve SAT problems. Even if is restricted to CNF, the current fastest algorithm CW to solve SAT satisfies as , and therefore it is true for more general circuits such as NC circuits or general polynomialsize Boolean circuits. As is shown in Lemma 1 below, Conjecture 3 can be based on SETH (CNF). Conjecture 2, which can be based on NCSETH AHVWW16 , can be considered as more stable than Conjecture 3, since NC circuits are more general than CNF. Furthermore, Conjecture 1 can be considered as more stable than Conjecture 2, because general polynomialsize Boolean circuits are more general than NC circuits.As is shown in Sec. III, Conjecture 1 prohibits only subexponentialtime classical sampling for the DQC1 model (Theorem 1) and the HC1Q model (Theorem 2). Conjecture 2, on the other hand, prohibits exponentialtime sampling for the DQC1 model (Theorem 3). Conjecture 3 with also prohibits exponentialtime sampling both for the DQC1 model (Theorem 4) and the HC1Q model (Theorem 5).
Lemma 1
For any CNF with , there exists a classical reversible circuit that uses only TOFFOLI and NOT such that , and for any .
Its proof is given in Sec. V.
In order to study Clifford quantum computing, we also introduce the following two conjectures.
Conjecture 4
Any (classical) deterministic algorithm that decides whether or for given (a description of) a 3CNF with clauses, , needs time. Here,
Conjecture 5
Any nondeterministic algorithm that decides whether or for given (a description of) a 3CNF with clauses, , needs time. Here
Iii Results
Based on conjectures introduced in the previous section, we now show finegrained quantum supremacy results. Proofs of the following theorems are given in Sec. V.
First, with Conjecture 1, we can show the following two results.
Theorem 1
Assume that Conjecture 1 is true. Then for any constant and for infinitely many , there exists an qubit DQC1 model, where , whose acceptance probability cannot be classically sampled in time within a multiplicative error .
Theorem 2
Assume that Conjecture 1 is true. Then for any constant and for infinitely many , there exists an qubit HC1Q model, where , whose acceptance probability cannot be classically sampled in time within a multiplicative error .
Note that although we know , we do not know the degree of the polynomial. For example, might be , and in this case, theorems say that the qubit DQC1 model and the qubit HC1Q model cannot be classically sampled in time, which is superpolynomial (subexponential) but not exponential.
Second, with Conjecture 2, we can show the following result.
Theorem 3
Assume that Conjecture 2 is true. Then for any constant and for infinitely many , there exists an qubit DQC1 model whose acceptance probability cannot be classically sampled in time within a multiplicative error .
Finally, with Conjecture 3, we can show the following two results.
Theorem 4
Assume that Conjecture 3 is true for . Then for any constant and for infinitely many , there exists an qubit DQC1 model whose acceptance probability cannot be classically sampled in time within a multiplicative error .
Theorem 5
Assume that Conjecture 3 is true for . Then for any constant and for infinitely many , there exists an qubit HC1Q model whose acceptance probability cannot be classically sampled in time within a multiplicative error .
The above three theorems, Theorem 3, Theorem 4, and Theorem 5, prohibit exponentialtime sampling of the DQC1 model and the HC1Q model.
For Clifford quantum computing, we can show the following two results. (Theorem 6 was also shown in Ref. Huang2 independently.)
Theorem 6
Assume that Conjecture 4 is true. Then for infinitely many there exists Clifford quantum circuit with gates whose acceptance probability cannot be calculated in time within an additive error smaller than .
Theorem 7
Assume that Conjecture 5 is true. Then for infinitely many there exists a Clifford quantum circuit with gates whose acceptance probability cannot be classically sampled in time within a multiplicative error .
For the PBC, we show the following result.
Theorem 8
Assume that Conjecture 5 is true. Then, for infinitely many there exists a qubit PBC whose output probability distributions cannot be classically sampled in time within a multiplicative error .
Iv Discussion
In this section, we give several discussions.
iv.1 Optimality of Theorem 5
The acceptance probability of any qubit HC1Q model can be calculated in time. Theorem 5 is therefore optimal. In fact, by a straightforward calculation,
for any reversible circuit and , where is the th bit of . If is size, each term of the exponential sum can be computed in time, and to sum all of them needs time. The total time is therefore
iv.2 and classical gates
When we consider quantum computing over Clifford and gates, we are interested in the number of gates, because gates are “quantum resources”. In a similar way, when we consider quantum computing over classical gates and , we are interested in the number of gates. Since the number of in the qubit HC1Q model is , Theorem 5 leads to the following corollary.
Corollary 1
Assume that Conjecture 3 is true for . Then for any constant and for infinitely many , there exists a quantum circuit with classical gates and gates whose output probability distributions cannot be classically sampled in time within a multiplicative error .
Actually, we can show a similar result based on Conjecture 1:
Theorem 9
Assume that Conjecture 1 is true. Then for any constant and for infinitely many , there exists a quantum circuit with classical gates and gates whose output probability distributions cannot be classically sampled in time within a multiplicative error .
Its proof is given in Sec. V.
For the strong simulation, we can show the following.
Theorem 10
Assume that SETH is true. Then, for any constant and for infinitely many , there exists an qubit quantum circuit over classical gates and gates such that and the classical exact calculation of cannot be done in deterministic time.
Its proof is given in Sec. V. Here, SETH asserts as follows.
Conjecture 6 (Seth)
Let be any classical deterministic time algorithm such that the following holds: given (a description of) a CNF, , with at most clauses, accepts if and rejects if , where
Then, for any constant , there exists a constant such that holds for infinitely many .
iv.3 Finegrained supremacy for the IQP model
Ref. Dalzell showed a finegrained result on the hardness of classically sampling output probability distributions of the IQP model within a multiplicative error. Here, the IQP model is defined as follows IQP1 .
Definition 6
The qubit IQP model with a unitary is the following quantum computing model, where consists of only diagonal gates (such as , , , and , etc.).

The initial state is .

The operation is applied on the initial state.

All qubits are measured in the computational basis.
To show the finegrained quantum supremacy of the IQP model, they used the conjecture, socalled poly3NSETH, which is the same as Conjecture 1 except that is restricted to be polynomials over the field with degree at most 3. An advantage of restricting to be polynomials is that IQP circuits can calculate polynomials over without introducing any ancilla qubit. (For the QAOA model, ancilla qubits are necessary Dalzell .) Because of this advantage, exponentialtime classical sampling is prohibited for the IQP model. However, a disadvantage of poly3NSETH is that it is violated when Tamaki . It was argued in Ref. Dalzell that improving the algorithm of Ref. Tamaki will not rule out poly3NSETH with , and therefore they conjecture poly3NSETH for . Since general Boolean circuits cannot be efficiently represented by systems of equations of polynomials, the technique of Ref. Tamaki cannot be used for general Boolean circuits of Conjecture 1.
iv.4 , , and gates
Strong simulation of the IQP is possible for by reducing the problem to counting the number of solutions of polynomials Tamaki . By using a similar technique, we can show the following.
Theorem 11
Let be an qubit size quantum circuit over , , and . For any , can be exactly calculated in deterministic time, where is the number of gates in .
It proof is given in Sec. V.
iv.5 Finegrained supremacy of the Boson sampling model
Ref. Dalzell also studied finegrained quantum supremacy of the Boson Sampling model. They introduced the conjecture, socalled perintNSETH, which states that deciding whether the permanent of a given integer matrix is nonzero needs nondeterministic time. In this case, no value of is ruled out by known algorithms Dalzell . It is not clear how perintNSETH and our conjectures are related with each other. At least we can show by using Ryser’s formula and Chinese remainder theorem that if of variable Boolean circuits are calculated in time , then permanents of integer matrices are calculated in time.
iv.6 Restricting to CNF
In Conjecture 1, we have assumed that is any polynomialsize Boolean circuit. It is still reasonable to consider Conjecture 1 with restricting to be CNF formulas while keeping the condition. (In fact, at this moment, the only known way of deciding whether or is to solve problems. The current fastest algorithm CW to solve of CNF does not contradict to the condition .) In this case, required quantum circuits should be simpler than those for general polynomialsize Boolean circuits.
iv.7 Nseth
Conjectures of the present paper and poly3NSETH of Ref. Dalzell are finegrained versions of . It is interesting to ask whether we can use NSETH CGIMPS , which is a faingrained version of , to show finegrained quantum supremacy. Here, NSETH is defined as follows.
Conjecture 7 (Nseth)
Let be any nondeterministic time algorithm such that the following holds: given (a description of) a polynomialsize Boolean circuit , accepts if and rejects if , where
Then, for any constant , holds for infinitely many .
At this moment, we do not know whether we can show any finegrained quantum supremacy result under NSETH. At least, we can show that proofs of our theorems (and those of Ref. Dalzell ) cannot be directly applied to the case of NSETH. To see it, let us consider the following “proof”. (For details, see Sec. V.) Given a Boolean circuit , we first construct an qubit quantum circuit such that if , and if , where and . By using Lemma 3, we next construct the qubit DQC1 model whose acceptance probability is
Then, if we assume that is classically sampled within a multiplicative error and in time, then NSETH is violated.
This “proof” seems to work, but actually we do not know how to construct such . In fact, the following lemma suggests that we cannot construct such .
Lemma 2
If such exists, then .
We do not know whether our conjectures can be reduced to more standard ones, such as SETH and NSETH. At least, we can show that Conjecture 1 is reduced to UNSETH (Unique NSETH) that is equal to NSETH (Conjecture 7) except that is promised for the no case. It means that if UNSETH is true, then Conjecture 1 is also true. In fact, for a given polynomialsize Boolean circuit , define the polynomialsize Boolean circuit by
for any . Then,
and therefore if then and if then .
iv.8 Other conjectures
In addition to SETH, NCSETH, and NSETH, there exists another conjecture, SETH, which asserts that for any there exists a large integer such that CNFSAT cannot be computed in time ParitySETH . Here, CNFSAT is the problem of computing the number of satisfying assignments of a given CNF formula modulo two. It is interesting to study whether we can find any finegrained quantum supremacy based on SETH. It is also open whether we can show any finegrained quantum supremacy under other conjectures that are not based on SAT, such as 3SUM 3SUM and AllPairs Shortest Paths problem (APSP) APSP .
iv.9 Additive error sampling
In this paper we have considered multiplicative error sampling. It is known that output probability distributions of several subuniversal quantum computing models, such as the Boson Sampling model BS , the IQP model IQP2 , the random gate model random , and the DQC1 model M , cannot be classically sampled in polynomial time within an additive error unless the polynomialtime hierarchy collapses to the third level. (In addition, two additional conjectures, socalled the averagecasehardness conjecture and the anticoncentration conjecture, are required for the Boson Sampling model. For the IQP model, the random gate model, and the DQC1 model, the anticoncentration conjecture is not a conjecture but a proven lemma.) Here, additive error sampling is defined as follows.
Definition 7
We say that a probability distribution is classically sampled in time within an additive error if there exists a classical probabilistic algorithm that runs in time such that
where is the probability that the classical algorithm outputs .
It is an important open question whether any finegrained version of those additiveerror results is possible or not.
iv.10 Stabilizer rank
We can also show lowerbounds of the stabilizer rank BravyiSmithSmolin , which is defined as follows.
Definition 8 (Stabilizer rank)
The stabilizer rank of an qubit pure state is the smallest integer such that can be written as
(1) 
where each is a coefficient and each is a Clifford circuit.
Note that the original definition of the stabilizer rank (Definition 8) does not care about computational complexity of and : the minimum of is taken over all decompositions of in the form of Eq. (1). In this paper, however, we consider only decompositions in the form of Eq. (1) such that there exists a time classical deterministic algorithm that, on input , outputs and a classical description of . Such an additional restriction is relevant when we study the stabilizer rank in the context of classical simulations of quantum computing.
The stabilizer rank is directly connected to the time complexity of classical simulations of quantum computing. For example, by using the wellknown gadget
where
is a magic state, we can easily show that for any universal quantum circuit that uses Clifford gates and gates, there exists a Clifford circuit such that
Since
and each can be computed in time, the value can be calculated in time (assuming that there exists a classical time algorithm that, on input , outputs and classical description of ). In this way, the stabilizer rank is directly connected to the time complexity of classical simulations. We do not know how to calculate the exact value of the stabilizer rank, and therefore finding better upperbounds of the stabilizer rank is essential. Several nontrivial upperbounds are known BravyiSmithSmolin , such as , which means
It is open how much can be we improve this upperbound. If we believe , it is clear that is impossible. It was conjectured in Ref. BravyiSmithSmolin that . Only known lowerbound is the very weak one BravyiSmithSmolin
which is not enough to show the conjecture.
Based on Conjecture 4 (ETH), we can show the following.
Theorem 12
Assume that Conjecture 4 is true. Let be a resource of Toffoli gates. Then, .
Its proof is given in Sec. V. Here, the meaning of the statement “ is a resource of Toffoli gates” is defined as follows.
Definition 9
Let be a nonClifford gate (such as , , or ). We say that an qubit state is a resource of gates if the following three conditions are all satisfied.

.

For any qubit quantum circuit over Clifford gates and gates, there exists an qubit Clifford circuit such that

The quantity
is computable in time.
For example, it is easy to verify that is the resource of gates.
In particular, if we take
where is the resource of a single gate, in Theorem 12, we obtain the following corollary.
Corollary 2
V Proofs
In this section, we provide proofs postponed.
v.1 Proof of Lemma 1
A CNF consists of AND, OR, and NOT, where
for any . An AND gate can be simulated by a TOFFOLI gate by using a single ancilla bit initialized to 0 (Fig. 1, left). An OR gate can be simulated by a TOFFOLI gate and NOT gates by using a single ancilla bit initialized to 0 (Fig. 1, right).
Let us define the counter operator by
where and . The counter operator can be constructed with generalized TOFFOLI gates. For example, the construction for is given in Fig. 2. It is clear from the induction that for general is constructed in a similar way. Each generalized TOFFOLI gate can be decomposed as a linear number of TOFFOLI gates with a single uninitialized ancilla bit that can be reused Barenco , and therefore a single requires a single uninitialized ancilla bit.
By using the counter operators, let us construct the circuit of Fig 3, which computes the CNF,
For a CNF, , it is clear from the figure that

ancilla bits initialized to 0 are necessary to calculate the value of a single clause. However, since these ancilla bits are reset to 0 after evaluating a clause, these ancilla bits are reusable.

To count the number of clauses that is 1, ancilla bits are necessary, where is the number of clauses. Note that , because

Each counter operator needs a single uninitialized ancilla bit. Since it is reusable, only a single ancilla bit is enough throughout the computation. This ancilla bit can also be used for the final bit TOFFOLI.

Finally, a single ancilla bit that encodes is necessary.
Hence, in total, the number of ancilla bits required is
v.2 Proof of Theorem 1
Let be (a description of) a polynomialsize Boolean circuit. Let be the number of AND and OR gates in . Since is polynomialsize, . Then by simulating each AND and OR in with TOFFOLI and NOT, we can construct the qubit unitary operator that uses only and TOFFOLI such that
for any , where is a certain bit string whose detail is irrelevant here. Define the qubit unitary operator by
Then, it is clear that
If then . If then . From Lemma 3 given below, by taking , we can construct the qubit DQC1 model such that its acceptance probability is
If then . If then . An qubit TOFFOLI can be decomposed into a linear number of TOFFOLI gates with a single ancilla qubit Barenco . In the construction, the ancilla qubit is not necessarily initialized, and therefore the completelymixed state can be used. Hence the qubit DQC1 model can be simulated by the qubit DQC1 model.
Assume that there exists a classical probabilistic algorithm that samples within a multiplicative error and in time . It means that
where is the acceptance probability of the classical algorithm. If then
and if then
It means that there exists a nondeterministic algorithm running in time such that if then accepts and if then rejects. However, it contradicts to Conjecture 1.
Lemma 3
v.3 Proof of Theorem 2
For given (a description of) a Boolean circuit , we again construct the qubit unitary operator such that
for any , where and is a bit string. Note that uses only and TOFFOLI. Consider the qubit HC1Q circuit in Fig. 5. By a straightforward calculation, the probability of obtaining the result , which we define as the acceptance probability , is
Then, if , . If , . The qubit TOFFOLI used in the circuit of Fig. 5 can be decomposed into a linear number of TOFFOLI gates with a single uninitialized ancilla qubit, which can be state Barenco . Therefore, the qubit HC1Q model is simulated by the qubit HC1Q model.
Assume that there exists a classical probabilistic algorithm that samples in time and within a multiplicative error :
where is the acceptance probability of the classical algorithm. Then, if ,
and if ,
It means that deciding or can be done in nondeterministic time, which contradicts to Conjecture 1.
v.4 Proof of Theorem 3
It is known that Cosentino any logarithmic depth Boolean circuit (that consists of AND, OR, and NOT) can be implemented with a polynomialsize quantum circuit acting on qubits such that
for all and , where is a certain function. Let us define the qubit unitary by
Then,
From now on the same proof holds as the proof of Theorem 1 with . Therefore, we can construct the qubit DQC1 model with such that its acceptance probability satisfies when , and when . If is classically sampled within a multiplicative error in time, Conjecture 2 is violated.
v.5 Proof of Theorem 4
For a given polynomialsize classical reversible circuit that consists of only NOT and TOFFOLI, its quantum version, , works as
for any , where is a bit string (it is actually the first bits of .) Therefore, the same proof as that of Theorem 1 holds by considering as . Hence we can construct the qubit DQC1 model with whose acceptance probability cannot be classically sampled within a multiplicative error in
Comments
There are no comments yet.