Quantum computation is widely believed to be intractable by classical computers. However, there also exist certain types of quantum circuits that can be efficiently simulated classically despite being able to generate highly entangled states. Famous examples are quantum circuits which consist of Clifford gates  or matchgates, corresponding to free-fermionic dynamics [37, 36, 8, 23]. Such classically simulatable quantum circuits are of importance because they illustrate what makes universal quantum computation different from classical computers. Moreover, they have practical applications such as randomized benchmarking [25, 26, 21], simulation by stabilizer sampling 
, and estimation of an error threshold of a quantum error correction code.
Classically simulatable or exactly solvable quantum circuits are also important in the study of dynamics of isolated quantum systems [32, 15, 13]. For example, Clifford circuits include quantum dynamics which are both integrable and non-integrable 111Throughout this paper, we say that quantum dynamics are integrable if there exist an extensive number of conserved quantities. , and they have been investigated from the perspective of quantum thermodynamics, and especially thermalization [18, 3]. Moreover, physical quantities, such as entanglement entropy and out-of-time-ordered correlators, of an ensemble of one and higher dimensional Haar random unitary circuits are calculated exactly [28, 29, 38]. Because, in general, it is notoriously difficult to treat quantum dynamics analytically except for one-dimensional integrable systems , above quantum circuit representations of quantum dynamics are powerful tools to analyze their physical properties. In addition, since two-dimensional quantum dynamics are not explored in comparison with one-dimensional cases, solvable models in two dimensions are in great demand.
Recently, a new class of quantum gates called “dual-unitary gates” has been introduced . Dual-unitary gates are unitary gates which remain unitarity under reshuffling their indices. The dynamics consisting of dual-unitary gates can be either integrable or non-integrable. In Refs. [4, 5], it has been shown that dynamical correlation functions and time evolution of operator entanglement entropy under dual-unitary quantum circuits (DUQCs) are calculated exactly. Because these quantum circuits have one spatial dimension, we call them one-dimensional (1D) DUQCs. Moreover, it has been shown that when the system size is infinite, time evolution of local observables, correlation functions, and entanglement entropy of 1D DUQCs arising from certain initial states can be calculated exactly .
Interestingly, despite the above property, dual-unitary gates contain arbitrary single-qubit gates and a certain class of two-qubit entangling gates. Note that these gates form a universal gate set if we can apply them freely. Nevertheless, the carefully constructed initial states and the infinite size limit allow us to compute the expectation values efficiently. Here, we ask whether they are classically simulatable or allows universal quantum computation if the system size is finite. The finiteness of the system size enables us to consider the circuit depth which scales with the system size and characterize their computational power.
In this paper, we investigate quantum computational power of 1D and two-dimensional (2D) DUQCs. Specifically, we characterize the computational complexity of the problem of calculating expectation values of local observables and the sampling problem of 1D and 2D DUQCs with finite system sizes. Additionally, we study classical simulatability of correlation functions of 2D DUQCs. Note that the generalization to the 2D lattice is of interest not only from the viewpoint of quantum computing but also from the viewpoint that exactly solvable quantum dynamics in two spatial dimensions is limited.
Our results are summarized in Fig. 1.
For the first problem, we show that expectation values of local operators of 1D and 2D DUQCs are exponentially close to Tr() until time, or circuit depth, and , respectively, where is a system length. In other words, local expectation values do not depend on specific choices of dual-unitary gates in those time regions, and they are classically simulatable. On the other hand, in later time, local expectation values of DUQCs can depend on dual-unitary gates, and we find that DUQCs can simulate universal quantum computation after time poly(). It means that the problem becomes -complete after time poly(). This contrasts to conventional classically simulatable quantum circuits with a fixed gate set, such as Clifford or matchgate circuits, where classical simulatability does not change depending on the circuit depth.
In addition, we show that sampling of the output of 1D and 2D DUQCs is intractable for classical computers after time and
, respectively, unless polynomial hierarchy (PH) collapses to its third level. This result is based on the fact that if a quantum circuit with post-selection can simulate universal quantum computation, the output probability distribution cannot be sampled by a classical computer efficiently unless PH collapses to its third level. It implies that, especially in the case of two dimensions, the sampling problem of constant-depth DUQCs is as hard as that of general constant-depth quantum circuits.
Moreover, we find that correlation functions of 2D DUQCs along a special direction become the trace of operators in linear depth, and hence they are classically simulatable. In contrast, the value of correlation functions along the other direction can depend on a choice of unitary gates, and they do not seem to be classically simulatable. We leave as an open problem whether or not the problem of calculating correlation functions of 2D DUQCs in linear depth is BQP-hard. Finally, we also show a sufficient condition of 2D lattices on which local observables of 2D DUQCs at an early time are classically simulatable. For instance, the lattices satisfying this condition include honeycomb lattices. In summary, we reveal that the computational power of DUQCs strongly depend on their circuit depth and problem settings. This provides a novel quantum computational model to investigate both classical simulatability of quantum computation and physical properties of non-equilibrium quantum systems.
The rest of the paper is organized as follows. In Sec. II, we introduce and review 1D DUQCs. In Sec. III, we characterize the complexity of the problem calculating local expectation values and the sampling problem of 1D DUQCs. In Sec. IV, we generalize 1D DUQCs to two spatial dimensions, and characterize the complexity as with the 1D case. After that, we discuss classical simulatability of correlation functions of 2D DUQCs and a generalization of lattices of qubits. Sec. V is devoted to conclusion and discussion.
Ii One-dimensional DUQCs
ii.1 Dual-unitary gates
We consider a -qubit system. Its computational basis is denoted by , where indicates a state of the -th qubit. A dual-unitary gate is a two-qubit gate in the following form:
where SWAP is the swap gate,
CZ is the controlled-Z gate,
, , and are arbitrary single-qubit gates, and both and are arbitrary real numbers. Alternatively, Eq. (1) is rewritten as
with , , , and . It has been shown that these gates can describe both integrable and non-integrable periodically driven quantum systems, or Floquet systems . For example, the time evolution operator of a periodically driven quantum system with XXZ interaction,
is one of the dual-unitary gates. The time evolution operator of a self-dual kicked Ising chain can also be written in terms of dual-unitary gates as follows:
where is the time ordered product, is the Dirac delta function, and is a real number.
Dual-unitary gates have the following nice property. Let be a nearest-neighbor two-qubit gate. Define , called the dual gate of , such that
Then, is a unitary gate if and only if
is a dual-unitary gate. This property can be expressed graphically by using a tensor-network representation of quantum circuits. We represent a two-qubit gateand as
where -legs and -legs of and serve as inputs and outputs, respectively. The unitarity, , can be written as
Similarly, the dual gate of and its Hermitian conjugate are represented as
where -legs and -legs of and serve as input and outputs, respectively. The property of a dual-unitary gate, namely , can be written as
ii.2 Dual-unitary quantum circuits
1D DUQCs are quantum circuits with qubits which consist of nearest-neighbor dual-unitary gates. They are defined as follows:
is an even number, and is a dual-unitary gate acting on qubit and at time , or graphically,
We note that dual-unitary gates can differ from each other, that is, the quantum dynamics can be inhomogeneous in space and time. In Eqs. (12) to (15), we assume a periodic boundary condition in space, that is, .
ii.3 Solvable initial states
In a 1D DUQC, when an initial state satisfies certain conditions and a system size is infinite, it has been shown that time evolution of local observables, correlation functions and entanglement entropy can be calculated exactly . Such an initial state is called a solvable initial state and can be described in terms of a matrix product state (MPS) .
Here we describe two conditions that make a two-site shift invariant MPS,
where is a -dimensional square matrix, a solvable initial state. can alternatively be represented by a tensor-network as,
Note that for large , which implies is needed in order to normalize , namely in the limit of . The second condition is that satisfies the following condition:
where is Kronecker’s delta. Eq. (21) implies
as right and left eigenvectors with an eigenvalue 1. Strictly speaking, the second condition considered here is more restrictive than that of Ref. . However, they are equivalent in the thermodynamic limit (see Theorem 1 in Ref. ), and we adopt Eq. (22) for clarity.
The simplest example of solvable states is a chain of EPR pairs , where , or graphically,
whose is zero. This example has been studied to analyze the entanglment dynamics of self-dual kicked Ising chains .
ii.4 Local expectation values of 1D DUQCs
Let us briefly describe how expectation values of local observables can be calculated for dual unitary circuits with solvable initial states . We consider a time-evolved transfer matrix defined as
is both right and left eigenvectors of with eigenvalue 1. In fact, is the unique eigenvector with maximum eigenvalue. This leads to the following equality:
By virtue of Eq. (28), one can calculate an expectation value of a local observable as follows:
where is a solvable initial state evolved to time . It means that an expectation value of a local observable does not depend on a specific choice of dual-unitary gates. From a similar argument, it has been also shown that correlation functions of 1D DUQCs are classically simulatable .
Iii computational power of 1D DUQC
iii.1 Local expectation values
Although local observables in a 1D DUQC are calculated exactly when the system size is infinite, a dual-unitary gate can contain arbitrary single-qubit gates and the CZ gate, which can form a universal gate set in principle . Thus, it is natural to ask whether or not DUQCs can perform universal quantum computation when their system size is finite. Finiteness of the system size enables us to consider the circuit depth which is scaling with the system size and characterize their computational power.
In this section, we answer the above question affirmatively and characterize computational complexity of the problem of calculating local expectation values for dual-unitary circuits. More precisely, we consider a local observable with length
where is an integer, and is an observable on qubit . Local expectation values of at time is . Here, we note that is not generally normalized for a finite . Then, we define the following decision problem, which has a parameter .
Problem 1 (local expectation values of 1D DUQCs).
Consider a 1D DUQC in time , a solvable -qubit initial state with , and a local observable whose operator norm is 1 with length . is promised to be either or , where . The problem is to determine whether is or .
We show that Problem 1 with time , for an arbitrary , is in . In contrast, with time , it becomes -complete. We first prove the above statements in the case where the initial state is a chain of EPR pairs Eq. (25), which is normalized for any , for simplicity. Then, we extend to general solvable initial states.
iii.1.1 1D DUQC is classically simulatable at an early time
First, we prove the former statement. In this case, an expectation value with time can be written as
where and are respectively assumed to be even for clarity, and the normalization coefficient which arises from Eq. (25) is omitted. To derive Eq. (31), we remove unitary gates outside of the causal-cone by using Eq. (9). Although we fixed parity of and , expectation values with other combinations of parity can be written likewise. If , we obtain by adapting Eq. (11) to the both sides of Eq. (31) sequentially and removing leftover unitary gates by adapting Eq. (11). In other words, an expectation value is identical to the expectation value of a maximally mixed state regardless of components of dynamics. The detail of the proof is in Appendix A. For , in general, the local expectation value is not equal to Tr as the dual-unitary gates cannot be canceled. However, local expectation values at time less than are classically simulatable because only constant number of unitary gates are involved in calculation of local expectation values.
We can easily generalize the above results to general solvable initial states. Let us note that a transfer matrix of a solvable MPS to the power can be written as
where is a matrix such that leading order of non-zero elements are and is the second largest eigenvalue of . By using Eq. (32), an expectation value of with can be calculated as follows:
where , deriving from the matrix , is . Then, by applying Eqs. (9), (11), (21), and (23) to Eq. (33), we obtain the expectation value . The detail of the calculation is in Appendix B. Because is exponentially suppressed with respect to , Problem 1 with is still in . Here, we note that if an initial state is a chain of EPR pairs, can be chosen as zero, that is, with is classically simulatable.
On the other hand, for time , the expectation value can be written as
In contrast to Eq. (31), we cannot use Eqs. (9) and (11) to calculate Eq. (34) efficiently. It could be possible that we can calculate Eq. (34) efficiently by another method. In the following section, we exclude the possibility by showing Problem 1 with is BQP-complete.
iii.1.2 1D DUQC is universal in late time
The inclusion of the problem in is trivial; we can just execute the dual-unitary circuit to obtain an expectation value. What is left to be shown is that the problem is -hard. To do that, we consider a -complete problem calculating whether an local expectation value is , where is promised to be either or , where , and is a 1D quantum circuit consisting of poly() nearest-neighbor two-qubit gates. Then, we show that Problem 1 is as hard as the -complete problem after time poly().
Firstly, we construct the CZ gate acting on an even numbered site and an odd numbered site by cancelling swap gates from a DUQC, as follows:
Secondly, arbitrary single-qubit gate can be implemented by substituting for the CZ gate in Eqs. (35) and (36). A quantum circuit consisting of arbitrary one-qubit gates and CZ gates has capability to efficiently simulate arbitrary quantum circuit consisting of poly() nearest-neighbor two-qubit gates . Therefore, we can construct a DUQC with poly() depth such that is equal to , which means that Problem 1 with time poly and input a chain of EPR pairs is a -complete problem.
iii.2 Sampling problem
We now move on to discuss the sampling complexity of 1D DUQCs. We show that classical simulation of sampling from linear depth 1D DUQCs is hard.
be the probability distribution withbeing the probability of obtaining output when is measured in computational basis. We define that a probability distribution is sampled by classical computers efficiently with a multiplicative error if there exists a classical probabilistic polynomial-time algorithm that outputs with probability such that , for all . It has been shown that if an -qubit quantum circuit with post-selection can simulate universal quantum computation, the measurement output of qubits cannot be sampled by classical computers efficiently with a multiplicative error unless polynomial hierarchy collapses to its third level . On the other hand, measuring a square lattice cluster state in an arbitrary measurement basis with post-selection can perform universal quantum computation . A square lattice cluster state is defined as
where is , we assigned a qubit in state to each vertex of a square lattice, and is the set of all edges of the lattice.
Here, we prove that a 1D DUQC can generate a square lattice cluster state with an EPR-chain initial state in time , which implies that sampling from is classically hard. We assume that is a square of an even number , which enables us to make a one-to-one correspondence between the 1D qubits and ones on square lattice. Note that the following equation holds:
where is the Hadamard gate. With this in mind, we obtain a square lattice cluster state by applying the following dual-unitary circuit to the initial state :
and at other odd and even times are and , respectively. Graphically, the above DUQC can be written as Fig. 3 (a), and it is equivalent to Fig. 3 (b). As shown in Fig. 3 (b) and (c), by rearranging the 1D qubit array (b) to the square lattice (c), the final state of the DUQC is equivalent to the square lattice cluster state. Moreover, because dual-unitary gates include arbitrary single-qubit gates, we can measure the final state in an arbitrary measurement basis. Therefore, sampling from depth- 1D DUQCs is unlikely to be classically simulatable.
Iv Generalization to 2D DUQCs
In this section, we generalize DUQCs to two spacial dimensions and characterize their computational power.
iv.1 Definition of 2D DUQCs
We consider a -qubit system on a square lattice. Its computational basis is denoted by , where indicates a state of the -th qubit.
We define , , , and as
where is a dual-unitary gate acting on qubit and qubit , and is an arbitrary two-qubit unitary gate acting on qubit and qubit . We also define as a matrix which is an arbitrary product of and , for example, , , or . We note that the fact that a unitary gate in or can be an arbitrary unitary gate is a significant difference between one and two spatial dimentions. In Eqs. (48) to (51), we assumed a periodic boundary condition in space, that is, and for all and . Then, we define 2D DUQCs are quantum circuits with qubits as follows:
where is a multiple of four and , , is a unitary at time .
In the following subsections, we consider initial states which are product states of solvable initial states aligned in rows:
where indicates the state of ()-th qubit. We call these initial states 2D solvable initial states in the sense that, as we show in the next section, local expectation values are classically simulatable at an early time.
The simplest example of 2D solvable initial states is rows of chains of EPR pairs
For concreteness, in this section, we fix and in Eq. (52) as and , respectively, but one can discuss the argument in this section similarly in the case of other and . Besides, as with 1D cases, dynamics also contain a 2D periodically-driven XXZ and a 2D self-dual kicked Ising model (The detail is shown in Appendix C.).
iv.2 Local expectation values
We characterize computational complexity of the problem of calculating local expectation values for 2D DUQCs. We consider a local observable in an square region
where is an observable of qubit for some integers and . Hereafter, for clarity, we assume , , and to be odd number, even number, and even number, respectively. Note that the discussion of the following subsections are not limited to this choce, and similar argument can be applied to other choice of parities. We denote by a 2D solvable initial state Eq. (54) evolved at time , that is, . Then, we define the following decision problems as with the 1D case.
Problem 2 (local expectation values of 2D DUQCs).
Consider a 2D DUQC in time with time t, a 2D solvable qubit initial state with (poly()), and a local observable whose operator norm is 1 with length . is promised to be either or , where The problem is to determine whether is or .
Similary to the 1D case, we show that Problem 2 is in with time , for , and -complete with time .
iv.2.1 2D DUQC is classically simulatable at an early time
The way to calculate local expectation values is similar to the 1D case except for contractions of unitary gates at even time. The procedure is to contract dual-unitary gates at odd time by using Eq. (9) and unitary gates at even time by using Eq. (11). As a result, if and , we obtain
Origins of the conditions are the same as those of 1D cases. We explain the procedure graphically in the case that an initial state is rows of EPR pairs (Fig. 4).
Firstly, we remove unitary gates out-side of the causal-cone. Then, the local expectation value can be represented as Fig. 4 (a). A tensor network in dotted line of Fig. 4 (a) is depicted in further detail in Fig. 4 (b). In the first equality in Fig. 4 (b), we use the definition of dual-unitary gates Eq. (11), and remove them. After that, unitary gates, which are contracted with removed dual-unitary gates (unitary gates painted in red and orange in Fig. 4), can be removed by the definition of unitary gates Eq. (9). This leads to the second equation in (b). By repeating this procedure, one obtain that local expectation values are . Moreover, local expectation values at time less than are classically simulatable because only constant number of unitary gates are involved in calculation of local expectation values. One can straightforwardly generalize the above results to general 2D solvable initial states similar to the 1D case and Appendix B. Altogether, we obtain Problem 2 with is in . Note that this is true even if and are replaced by . For example, in the case of , dual-unitary gates and unitary gates are contracted and removed as Fig. 4 (c).
We note that classical simulatability of 2D DUQC are also interesting as a solvable model of quantum dynamics because analytical research on quantum dynamics in 2D systems is much more difficult than 1D cases. In fact, all local expectation values of 2D DUQCs become Tr() in the thermodynamic limit, which means that a local density matrix of a 2D solvable initial state evolved by 2D DUQCs is identical to a thermal equilibrium state at infinite temperature. Therefore, thermalization of solvable initial states can be shown analytically in the thermodynamic limit. Understanding conditions when thermalization happens is one of the most important problems in non-equilibrium physics [32, 15, 13], and it means that 2D DUQCs are rare toy models of 2D non-equilibrium quantum physics.
iv.2.2 2D DUQC is universal in late time
iv.3 Sampling problem
We now move on to discuss the sampling complexity of 2D DUQCs. We show that a constant depth 2D DUQC can generate a square lattice cluster state.
We obtain a square lattice cluster state by four depth 2D DUQCs as follows:
Therefore, sampling the output of constant depth 2D DUQCs is unlikely to be classically simulatable.
iv.4 Correlation functions
Let us discuss classical simulatability of two-point correlation functions. For simplicity, we assume that the initial state is , but one can easily generalize the following argument to general solvable initial states. In such a case, we expect two-point correlation functions to be anisotropic because solvable initial states are anisotropic. We consider two types of correlation functions, one of which is classically simulatable, and the other does not seem to be classically simulatable.
First, we consider the following correlation function, which is classically simulatable:
where is evolved at time , and is an observable of qubit (). When holds, , which can be graphically represented as Fig. 5 (a), can be straightforwardly calculated as with calculation of expectation values shown in Fig. 4. As a result, we have In other words, qubit and qubit cannot be correlated for an arbitrary 2D DUQC in the time region. Second, we consider the following correlation function, which is expected to be classically intractable:
We also assume that holds. can take a nonzero value only in the case of and odd , , or and even . In those cases, we conjecture that is unlikely to be classically simulatable because of the following reason. can be graphically represented as Fig. 5 (b). After contracting unitary and dual-unitary gates of Fig. 5 (b) by using Eq. (9) and Eq. (11), the remaining gates of it can be represented as Fig. 5 (c). This is similar to correlation functions of 1D DUQCs (see Fig. 3 of Ref. ), but important difference is that uncontracted gates form a 2D tensor-network in 2D DUQCs. Because of this, calculating correlation functions in 2D DUQCs seems to be hard for a classical computer. Therefore, classical simulatability of correlation functions seems to depend on the relative position of two local observables. It is still an open problem whether or not calculating with the condition is -hard.
iv.5 General lattice pattern
So far, we only consider dynamics in a 2D square lattice. It is natural to consider a generalization to other lattices, such as a honeycomb lattice. We note that unitaries at even-time of dynamics defined in subsection IV.1 can be chosen as identity gates. If a unitary gate is an identity gate at all time, the edge between qubit and qubit can be effectively removed. So, if we construct lattices by eliminating edges in -direction, 2D DUQCs with such a lattice can be treated as with ones with a 2D square lattice. We name such lattices solvable lattices. Such lattices include, for example, a honeycomb lattice. This is illustrated by Fig. 6. One can construct an arbitrary solvable lattice in the same way. Therefore, local expectation values and correlation functions of 2D DUQCs with solvable lattices are classically simulatable at an early time as same as those with square lattices.
V Conclusion and discussion
We have investigated computational complexity of the problems calculating physical properties of 1D and 2D DUQCs. First, we have shown that the complexity of calculating local expectation values of dual-unitary quantum circuits highly depends on their circuit depth. Second, we have shown that classical simulation of sampling from 1D and 2D DUQCs after linear and constant depth, respectively, is hard.
The first result is in contrast to conventional classically simulatable quantum circuits with fixed gate sets such as Clifford circuits and matchgates, whose classical simulatability does not change depending on the circuit depth. The dual-unitary quantum computational model is the first example of the model, where quantum computational power makes a transition between () time and poly(
) time. It is reminiscent of dynamical phase transitions of computational complexity which have been recently studied in other contexts[14, 27, 30]. It would be interesting to investigate whether or not local expectation values of DUQCs at an early time slightly later than that considered in this paper are also classically simulatable.
Besides, because analytic research on quantum dynamics in 2D systems is much more difficult than 1D cases, generalization of DUQC to two spatial dimensions are also interesting as a solvable model of quantum dynamics. It would be important to generalize DUQCs and solvable initial states to higher than two spatial dimensions. We expect that a high-dimensional DUQC will deepen our understanding of a non-equilibrium phenomenon in a high-dimensional quantum system, such as a 2D self-dual kicked Ising model.
This work is supported by MEXT Quantum Leap Flagship Program (MEXT QLEAP) Grant Number JPMXS0118067394 and JPMXS0120319794. KM is supported by JST PRESTO Grant No. JPMJPR2019 and JSPS KAKENHI Grant No. 20K22330. KF is supported by JSPS KAKENHI Grant No. 16H02211, JST ERATO JPMJER1601, and JST CREST JPMJCR1673.
Appendix A A calculation procedure for local expectation values of a chain of EPR pairs in 1D DUQCs
In this appendix, we show that local expectation values for :
Appendix B Local expectation values of general solvable initial states.
In this appendix, we show that local expectation values at time with general solvable states are approximated by Tr(). Local expectation values without normalization can be written as
where and are the total number of tensors and ones which are outside of the causal-cone, respectively. By inserting identity operators , one obtains
The second factor of Eq. (70) is equal to a matrix element of , where is the transfer matrix defined in Eq. (20). By the assumption that is the unique eigenvector of the matrix with the latgest eigenvalue , the Jordan canonical form of can be written as
where is the number of Jordan blocks, is the diagonal part, is the nilpotent part, and is less than and ordered in descending order. Let be . Then, the n-th power of a transfer matrix can be written as .
Now we evaluate local expectation values. First, in the same way as the calculation in Appendix A, we obtain that
is equal to Tr(), which arises from and is a dominant term of Eq. (70).
Next, we calculate the error term , which arises from . First, we evaluate the second factor of Eq. (70), namely, . We denote the
-norm of the vectorby and the operator norm of a matrix by . Because of
where is a matrix, the following holds:
By the binomial theorem, is expanded as
where is the dimension of the -th Jordan block, that is, the dimension of the space on which acts nontrivially, and we used the fact that
is the zero matrix. Then,is upper bounded as follows: