WO2025116818A1 - Methods and systems for decomposing a quantum circuit - Google Patents

Abstract

Methods and systems for decomposing a quantum circuit are described A method of decomposing a quantum circuit comprises: dividing the quantum circuit into a first sub- circuit and second sub-circuit; determining a pre-processing unitary operator and a post-processing unitary operator which approximate local evolutions on the first sub- circuit and second sub-circuit; determining gate decompositions of the pre-processing unitary operator and the post-processing unitary operator; and using the gate decompositions of the pre-processing unitary operator and the post-processing unitary operator to determine a gate decomposition of the quantum circuit.

Description

METHODS AND SYSTEMS FOR DECOMPOSING A QUANTUM CIRCUIT

TECHNICAL FIELD

The present disclosure relates to quantum computing and in particular to decomposing quantum circuits for circuit compilation.

BACKGROUND

Quantum computing promises immense computational benefits from solving classically intractable problems to efficiently simulating quantum interactions. Yet, such tasks typically require the synthesis of complex n-qubit unitary operations using elementary quantum gates. While systematic decomposition techniques exist, all such generic solutions require gate counts that scale exponentially with n. To reduce such intractable realizations into a shallow circuit - even when possible - remains a highly intractable (NP Complete) task. Thus, quantum circuit compilation, which is the capacity to design a simpler circuit for implementing a given unitary process, is a pressing task; motivated by recent variation algorithms for quantum circuit discovery.

SUMMARY

According to a first aspect of the present disclosure, a method of decomposing a quantum circuit is provided. The method comprises: dividing the quantum circuit into a first sub-circuit and second sub-circuit; determining a pre-processing unitary operator and a postprocessing unitary operator which approximate local evolutions on the first sub-circuit and second sub-circuit; determining gate decompositions of the pre-processing unitary operator and the post-processing unitary operator; and using the gate decompositions of the pre-processing unitary operator and the postprocessing unitary operator to determine a gate decomposition of the quantum circuit.

The method may further comprise decomposing the first sub-circuit and the second sub-circuit into smaller sub-circuits. The process may be repeated while limiting a gate depth of the quantum circuit. In an embodiment, determining gate decompositions of the pre-processing unitary operator and the postprocessing unitary operator comprises parametrizing the preprocessing unitary operator and the post-processing unitary operator and minimizing a cost function.

The cost function may be a global cost function or a sum of local cost functions each restricted to a respective qubit.

In an embodiment, determining gate decompositions of the pre-processing unitary operator and the post-processing unitary operator comprises a gradient descent method.

In an embodiment, determining decompositions of the pre-processing unitary operator and the post-processing unitary operator comprises controlling a quantum processing unit to input entangled qubit pairs into the quantum circuit and measuring a the cost function and its gradient using a destructive swap test. The entangled qubit pairs may be entangled via a CNOT gate. The destructive swap test may comprise applying CNOT and Hadamard gates.

According to a second aspect of the present disclosure, a computer readable medium storing processor executable instructions which when executed on a processor cause the processor to carry out a method as set out above is provided.

According to a third aspect of the present disclosure, a system for decomposing a quantum circuit is provided. The system is configured to: divide the quantum circuit into a first sub-circuit and second sub-circuit; determine a pre-processing unitary operator and a post-processing unitary operator which approximate local evolutions on the first sub-circuit and second sub-circuit; determine gate decompositions of the pre-processing unitary operator and the post-processing unitary operator; and use the gate decompositions of the pre-processing unitary operator and the post-processing unitary operator to determine a gate decomposition of the quantum circuit.

In an embodiment, the system is further configured to decompose the first sub-circuit and the second sub-circuit into smaller sub-circuits. In an embodiment, the system is further configured to limit a gate depth of the quantum circuit.

In an embodiment, the system is further configured to determine gate decompositions of the pre-processing unitary operator and the post-processing unitary operator by parametrizing the pre-processing unitary operator and the post-processing unitary operator and minimizing a cost function.

The cost function may be a global cost function or a sum of local cost functions each restricted to a respective qubit.

In an embodiment, the system is further configured to determine gate decompositions of the pre-processing unitary operator and the post-processing unitary operator by a gradient descent method.

In an embodiment, the system further comprises a quantum processing unit and is further configured to determine decompositions of the pre-processing unitary operator and the post-processing unitary operator by controlling the quantum processing unit to input entangled qubit pairs into the quantum circuit and measuring the cost function and its gradient using a destructive swap test. The entangled qubit pairs may be entangled via a CNOT gate. The destructive swap test may comprise applying CNOT and Hadamard gates.

BRIEF DESCRIPTION OF THE DRAWINGS

In the following, embodiments of the present invention will be described as non-limiting examples with reference to the accompanying drawings in which:

FIG.1 is a schematic diagram showing quantum circuit decoupling according to an embodiment of the present invention;

FIG.2 is block diagram showing a quantum circuit decomposition system according to an embodiment of the present invention; FIG.3 is a flow chart showing a method of quantum circuit decomposition according to an embodiment of the present invention;

FIG.4 is a schematic diagram showing a variational quantum decoupling algorithm according to embodiments of the present invention;

FIG.5A shows a general 2-bit unitary circuit and its decomposition;

FIG.5B is a graph showing results of variational compilation of the general 2-bit unitary circuit shown in FIG.5A;

FIG.6A shows a generic 4-bit unitary circuit;

FIG.6B is a graph showing results of variational compilation of the generic 4-bit unitary circuit shown in FIG.6A;

FIG.7A shows a 4-qubit unitary circuit that is generated by a spindle-shaped circuit;

FIG.7B is a graph showing results of variational compilation of the 4-bit unitary circuit shown in FIG.7A;

FIG.8 is a graph showing raw data for the results shown in FIG.7;

FIG.9 is a quantum circuit diagram showing a conventional swap test;

FIG.10 is a quantum circuit diagram showing a destructive swap test;

FIG.11 is a quantum circuit diagram showing a circuit that prepares Bell state; and

FIG.12 is a quantum circuit diagram showing the splitting of a parameterized quantum circuit into three parts.

DETAILED DESCRIPTION In the present disclosure variation circuit decoupling is proposed. Variational circuit decoupling is a divide-and-conquer approach that enhances variational circuit algorithms. The procedure involves designing a variational decoupling algorithm that first breaks a many-qubit interaction into smaller components. We can then optimize these smaller components individually or break them down further via recursive applications of variational decoupling. We demonstrate the method for compiling circuits approximating arbitrary two and four-qubit gates, where it identifies circuits of much higher fidelities than direct variational methods.

When studying a system of coupled harmonic oscillators, a key approach is to decouple its dynamics into various normal modes. We can then study each mode individually, enabling a deeper understanding of underlying dynamics. In the era of computer simulation, this divide-and-conquer approach has further operational importance, enabling leverage of parallel processing, and reducing the computational costs of simulation or optimization. Decoupling complex systems into simpler components has seen success in diverse settings, from modeling the motion of human hands to hydraulic simulation.

FIG.1 is a schematic diagram showing quantum circuit decoupling according to an embodiment of the present invention. As shown in FIG.1 , an unknown unitary quantum process U 100 is decoupled into smaller components.

We adopt the premise of variational quantum circuit discovery, where we are given black-box access to some unknown n-qubit unitary quantum process U. This could represent a complex quantum circuit we wish to simplify; some unknown quantum device we wish to reverse-engineer, or some natural quantum process we wish to simulate. Our goal is to approximate this process with a sequence of elementary quantum gates - subject perhaps to constraints on circuit complexity or depth. Systematic solutions to this problem are clearly intractable. Process tomography to determine the matrix elements of U alone would scale exponentially with n, as would any general method of decomposing U into elementary gates. Quantum-aided variational approaches aid in accelerating the process by use of quantum computers themselves. They typically involve two components: (a) An ansatz, a family of quantum circuits W(6) parameterized by some k-dimensional vector 0 = (9o, . . . , 9k-i), and (b) A cost function C that that measures the performance loss of each candidate circuit in approximating W. The goal is to identify the parameter set 0 so that the corresponding circuit W(0) has a minimal cost. 0 corresponds to easily adjustable parameters of some proposed circuit architectures (e.g. magnitude of rotations on various single-qubit gates), such that knowledge of 6 allows the physical synthesis of the associated quantum circuit W(9). Meanwhile, the cost function is generally chosen to be normalized (takes a maximum value of 1 ) with the following properties:

Measurable and Gradient Measurable, such that C(W) and each partial derivative ^-C can be efficiently estimated, i.e., they each can be determined to target precision

E using at most poly(n, 1/ e) calls to IV and classical computing costs that scale efficiently with n.

Faithful, such that C is non-negative, and C(W) = 0 if and only if no general n-qubit quantum circuit are preferred over HZ.

For example, 1 - F(u, f/)is a natural candidate for the cost function, where F(u, U~)\s the gate fidelity of the synthesize unitary U with respect to the target unitary U. This then allows gradient descent-based methods to find 9 within the parameterized circuit to minimize the cost. However, the rate at which variational circuits converge can become severely limited due to barren plateaus especially in scenarios where circuits being optimized over had no constraints.

As shown in FIG.1 , given a target complex unitary quantum operation U, the decoupling of U involves identifying shallow pre and postprocessing operation Vo and Vi, such that U ~ VI(UA ® UB)VO, for some operators UA and UB which act locally on subsystems A and B respectively. This procedure can be repeated recursively, breaking down UA and UB into smaller components, until all subsystems are sufficiently small. Take a 4-qubit system for example, the target unitary operator U is first decomposed by Vo and Vi into local operators on subsystems A, B, and the local operator UA (UB) is further decomposed by VXoand VA,I (VB,O and VB,I) into single qubit unitary operators (UAA, UAB, UBA, and UBB).

FIG.2 is block diagram showing a quantum circuit decomposition system according to an embodiment of the present invention. The quantum circuit decomposition system 200 is a computer system with memory that stores computer program modules which implement methods of decomposing a quantum circuit according to embodiments of the present invention.

The quantum circuit decomposition system 200 comprises a processor 210, a working memory 212, a user interface 214, a network interface 216, a quantum processing unit interface 218, program storage 220, and a quantum processing unit 230. The processor 210 may be implemented as one or more central processing unit (CPU) chips. The program storage 220 is a non-volatile storage device such as a hard disk drive which stores computer program modules. The computer program modules are loaded into the working memory 212 for execution by the processor 210. The user interface 214 is an interface which allows data and controls to be input by a user and also allows the user to receive outputs such as the results of methods of decomposing a quantum circuit. The network interface 216 is a wired or wireless network interface which allows the quantum circuit decomposition system 200 to send an receive data over a network such as the internet. The quantum processing unit interface 218 controls the quantum processing unit 230. The quantum processing unit 230 allows the quantum circuit decomposition system 200 to manipulate qubits to carryout quantum processing. As shown in FIG.2, the quantum processing unit 230 carries out quantum processing on the quantum circuit 100 which is to decomposed using methods described in the present disclosure. The quantum processing unit 230 may be configured to allow quantum circuits to be attached such that the qubits input into the quantum circuit and output from the quantum circuit can be generated and processed by the quantum processing unit 230.

The program storage 220 stores a quantum circuit decomposition program 222. The quantum circuit decomposition program 222 causes the processor 210 to execute various methods of decomposing a quantum circuit which are described in more detail below. The program storage 220 may be referred to in some contexts as computer readable storage media and/or non-transitory computer readable media. As depicted in FIG.2, the quantum circuit decomposition program 222 is a distinct module which performs respective functions implemented by the quantum circuit decomposition system 200. It will be appreciated that the quantum circuit decomposition program 222 may be implemented as multiple modules and the boundaries between these modules are exemplary only, and that alternative embodiments may merge modules or impose an alternative decomposition of functionality of modules. For example, the program discussed herein may be decomposed into sub-modules to be executed as multiple computer processes, and, optionally, on multiple computers. Moreover, alternative embodiments may combine multiple instances of a particular module or sub-module. It will also be appreciated that, while a software implementation of the computer program modules is described herein, these may alternatively be implemented as one or more hardware modules (such as field-programmable gate array(s) or applicationspecific integrated circuit(s)) comprising circuitry which implements equivalent functionality to that implemented in software.

While the quantum circuit decomposition system 200 is described above as having both classical and quantum computational functionalities, it is envisaged that embodiments may be implemented using solely quantum computational functionalities.

FIG.3 is a flow chart showing a method of quantum circuit decomposition according to an embodiment of the present invention. The method 300 shown in FIG.3 is carried out by the quantum circuit decomposition system 200 shown in FIG.2.

In step 302, the quantum circuit 100 is divided into a first sub circuit and a second sub circuit. We adopt a divide-and conquer approach. Instead of attempting to learn a circuit decomposition for the entirety of U, we break the circuit down into a series of smaller circuits as illustrated in FIG.1. Specifically, we first divide n-qubit system into two subsystems A and 8 of approximately equal size (i.e., of size [n/2] and [n/2]).

In step 304, pre-processing and post-processing matrices (Vo and Vi) are determined. We identify decomposition of pre-processing and post-processing unitary matrices such that Vh UV^o is most closely approximated by local unitary evolution on A and 8. That is, identify Vo and Vi such that there exists UA on A and UB on B in which VUUVto most closely approximates UA ® UB.

In step 306, the gate decomposition of the decomposition of pre-processing and postprocessing unitary matrices is determined. We document below a variational means to learn gate decomposition for Vo and Vi without needing any knowledge of unitary operators UA and UB. After which, we may repeat the decoupling method on each of the localized unitary gates, breaking each of them down into two smaller gates on « n/4 qubits (that is the method may return to step 302 in which each sub circuit is divided into smaller sub circuits). This can proceed until all localized operators are sufficiently small for conventional variational methods (e.g. a single qubit).

Finally in step 308, the various sub-problems are recombined to give a full circuit decomposition of U.

FIG.4 is a schematic diagram showing a variational quantum decoupling algorithm according to embodiments of the present invention. As shown in FIG.4, to decouple an n-qubit circuit 100, our algorithm begins with two systems of n/2-qubits - corresponding to the upper layer 402 and the lower layer 404 of the n-qubit circuit 100 shown in FIG.4. The two systems of n/2-qubits are labelled as a and /3.

Each layer is further divided into the 2 subsystems we wish to decouple, resulting in the 4 subsystems

of dimensions CIA and 0-C_{B a} H_B,p of dimensions CIB.

We apply W = VU UV^o to each of subsystem a and /3. The input state is initialized as

T_A®T_B where r_k - is the projector to the symmetric subspace of

® H_k p for k e {4, B}. This is done by first preparing each qubit in

_a and N_kp in a random computational basis and state, followed by qubit-wise application of the Hadamard gate for each /3-layer qubit. Each qubit is then entangled with a corresponding qubit in M_{k a} via a CNOT gate 406. Meanwhile, we can efficiently measure cost function CD and its gradient using a destructive swap test - which similarly involves only qubit-wise CNOT and Hadamard gates, followed by projective measurement 408. This information can then be used to tune 9o and 01 using techniques from variational circuits in the quantum circuit decomposition system 200. The key behind the variational decoupling algorithm is to find an appropriate cost function. First, let V e the parametrizations of the circuits K_o and V

_± with respective parameters 9

_{± 2} Meanwhile set W as the resulting

quantum process formed by pre and postprocessing of U. Since we can always engineer W given

and V_r, we can define our cost function concerning W alone for convenience. Here we desire a normalized cost function that (i) is faithful, such that

and (ii) exhibits both self and gradient measurability. Namely consider:

ntropy and the integration is taken over Haar random initial states \ip)_A on A and |0>_B on B. This cost function is faithful. That is C_sep(W) = 0 if and only if W the equivalent to the identity or the swap gate up to local unitary gates. Moreover, we develop efficient methods to measure both its value and its gradients.

To efficiently measure C_sep, we developed a method using a circuit with twice the size of W, illustrated in FIG.4. The initial stat

) in FIG.4 is defined as

where d_k is the dimension of the system

and P_s is the projector to the symmetric subspace of

This state can be prepared with a constant depth of quantum circuit with gate determinations efficiently achievable through a classical process. The estimation error of the cost function is bounded by the Hoeffding’s inequality since the cost function embodies the mean of a random variable based on the measurement outcomes. Using this technology we can then proceed with variational decoupling by choosing appropriate ansatzes

such that IV is parameterized by the joint vector 9* ==

® 0_X* . For each candidate VQ(0_O) and 14(01) > we can construct a realization of IV, and use above methods to efficiently measure C_sep and its various gradients, and thus employ gradient-descent-based optimization method to find suitable 7₀ and V₁ that decouples U. Note that, just as in standard circuit compilation, we may also choose to limit the gate depth of V_o and V_± should we wish for shallow circuit approximations of U.

Performance Comparisons

In the following section we demonstrate the improved efficacy of our decoupling method in two distinct scenarios:

The exact compilation of a general 2-qubit gate - where the ansatz is shown in FIG.5. Here, the circuit family is universal, and can theoretically synthesize any 2-qubit unitary operator. Our decoupling algorithm breaks this into two stages. The first decouples U into two single qubit circuits. The second stage then optimizes these circuits. Note that in this special case, we have set Vi = I, since this is already sufficient for compiling an arbitrary 2-qubit gate.

A shallow circuit approximation of an arbitrary 4-qubit gate, where the structure of the ansatz is shown in FIG.6. For both Vo and Vi, we utilize four layers of decoupling circuit, where each layer comprises a shallow circuit and single qubit and control gates. The circuit compilation is done via three stages. Stage one decouples the circuit into two generic 2-qubit gates, VA and VB. The second stage decouples these 2-qubit gates further into 4 single-qubit interactions. Finally, stage 3 then optimizes each of the four single-qubit gates.

FIG.5A shows a general unitary circuit and a FIG.5B is a graph showing results of variational compilation of the general 2-bit unitary circuit.

As shown in FIG.5A, the method divides the circuit 500 into single single-qubit unitary Operators 510. The division maximizes the fidelities. The graph in FIG.5B shows the fidelity F compared between the direct compilation (broken lines) and our decoupling method (solid lines). For each method, the training is repeated 20 times with different initializations. The data of each method are divided into quartiles, and the lines represent the median of each process. The second and third quartiles are shaded with the corresponding color. The preprocessing ansatz is chosen to be the universal two- qubit circuit, and the post-processing is set to identity. We see that the average fidelity does not increase during the decoupling stage, but greatly enhances training in the second stage by breaking up the problem into more tractable components. The resulting compiled circuits have significantly higher fidelity, such that 1 - F is reduced by a factor of 3 over the state of the art.

FIG.6A shows a generic 4-bit unitary circuit and FIG.6B is a graph showing results of variational compilation of the generic 4-bit unitary circuit.

As shown in FIG.6A, the optimization is divided into three phases. The circuit 600 is divided into 3 types: 4-qubit ansatzes 602, 2-qubit ansatzes 604, and the middle single qubit gates 606. Similar to the 2-qubit case, the data of each process is collected from 20 independent training processes. The layers are repeated for 4 times for Vo, Vi, and the layers in the 2-qubit ansatzes VA,O, VB,O, VA,I, VB,I repeat twice. The ansatzs are designed to explore stronger expressive power with a limited number of gates, which can be generalized to any number of qubits. In this case, our ansatz - being low depth - is non-universal. Nevertheless, we see that variation decoupling continues to achieve circuits of higher fidelity.

In each setting, we select target U at random according to the Haar measure. The former scenario represents the case where we have a universal circuit ansatz and which to exactly compile a unitary. The latter represents the scenario where we wish to use a shallow circuit with limited expressivity to approximate an arbitrary 4-qubit gate. We benchmark our method against standard variational approaches, where all circuit parameters are optimized by gradient descent to maximize the gate fidelity F simultaneously. In the standard approach, this optimization is done by minimizing a cost function CHST or the localized cost function CLHST. Here C_HST =

CHST ^are faithful measures that reach 0 when F approaches 1 . The C^_T are the cost function C_HST restricted to the /th qubit. For a fair comparison, all methods use the same circuit parameterization and thus have identical expressivity and all are trained using the ADAM method described in Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization, January 2017 with identical hyper-parameters. The results are shown in FIG.5B (2- qubit) and FIG.6B (4-qubit). In both cases, performance is gauged by how F(U_f U) improves, i.e., the fall of C_HST with the number of training iterations. We see that:

In 2-qubit gate decomposition, our decoupling approach converges a lot more quickly. For example, it achieves a fidelity of 0.9999 using the same amount of time where conventional methods approach a fidelity of 0.999.

In shallow circuit approximation of general 4-qubit gates, our decoupling method reaches circuit fidelity around 0.7, significantly ahead of conventional methods which achieves a fidelity of around 0.5. Note that high fidelity values are not expected here as our ansatz - being shallow - has limited expressivity.

FIG.7A, FIG.7B and FIG.8 show an analysis of a third scenario where target 4-qubit unitaries can be expressed by our shallow circuit.

FIG.7A shows a 4-qubit unitary circuit that is generated by a spindle-shaped circuit and FIG.7B is a graph showing results of variational compilation of the 4-bit unitary circuit. Similar to the ansatz described above with reference to FIG.6A, the circuit 700 is divided into 3 types: 4-qubit ansatzes 702, 2-qubit ansatzes 704, and the middle single qubit gates 706. However, the ansatzes Vo, Vi, and VA,O, VB,O, VA, T, VB, I are not repeated, and each of them only contains one layer of CNOT gates.

As can be seen from the graph in FIG.7B, if the 4-qubit unitary is generated by a circuit with a similar structure as our ansatz, our method can achieve a higher fidelity. FIG.8 is a graph showing raw data for the results shown in FIG.7B. Here our decoupling method continues can reach a circuit fidelity of 0.998 vs 0.9 in using conventional methods.

In all cases, conventional variational circuits exhibit steady gate fidelity gain till some saturation point. Our decoupling method, on the other hand, only minimizes CLHST in the final stage. Previous stages concentrate on decoupling the unitary and do not immediately improve gate fidelity. However, once decoupled, convergence to a low CHST occurs very quickly - more than compensating for the delayed start. Moreover, the final fidelity obtained shows marked improvement for both exact gate decomposition and shallow circuit approximation.

In the quantum theory, the states of a quantum system are described by the density operators p that act on a Hilbert space X, such that the operators p are positive semidefinite and tr[p] = 1. A linear map r from a system X to system Y is called a quantum channel if it maps a density operator in X to a density operator in Y. Such property of the linear maps is usually broken into two constraints: 1 . the map is completely positive (CP), and 2. the map is trace-preserving (TP). A map FJ -> Y is CP if, for all ancillary system

maps positive semidefinite operators p e X ® c/Z to positive semidefinite operators A map is TP if it preserves the trace of

operators, such that tr[r(p)] = tr[p] for all operators defined on X. For convenience, we usually investigate these two constraints with the matrix representations of the linear maps. The Choi-Jamiolkowski operator is one of the most commonly used representations of quantum channels. For each quantum channel T , its Choi- Jamiolkowski operator /^r is unique.

Definition 1. The Choi-Jamiolkowski operator J^r of a linear map

-» Y is defined as where is a copy of X, and |i) forms a basis of X.

The properties of CP and TP can be characterized easily with Choi-Jamilkowski operators: Theorem 1. A linear map r is a quantum channel if and only if J^r is positive semidefinite (

The Choi-Jamiolkowski operator also provides a convenient way to represent the input states and the measurements on the outputs of the quantum channels:

Theorem 2. Given quantum state p defined on X, an observable 0 defined on Y, and a quantum channel T: X -> Y, the expectation value of 0 with respect to r(p) is given by

^r

In the scope of this disclosure, we will focus on the endomorphisms of a system X, which are those linear maps T: X X whose output system is identical to the input system. More specifically, we will focus on the quantum channels that are induced by unitary operators, namely unitary channels. Let U denote the unitary channel with respect to a unitary operator U acts on X, such that [/(•) = U(/)Ut Then the Choi- Jamiolkowski operator of 'll is given by

which is a rank-one projector. For a given unitary U, there is a unique |I|J) (up to a global phase), such that J^u - |I|J) <I|J| . In the special case that U = I, the state is the unnormalized maximally entangled state |’P⁺) . According to the TP property of quantum channels, | I where d is the dimension of X.

Measurements and State Preparation

FIG.9 shows a conventional swap test circuit. It is known that the purity, Tr[p²], of a state p can be measured with a swap test efficiently. With the ability to perform a controlled swap between two copies of p, the value of Tr[p²] is proportional to the probability of measuring 0 in the circuit of FIG.9. Alternatively, the measurement of purity can be done with pair-wise Bell measurements instead of the controlled swap test. This is done by a bit-wise bell measurement on a pair of systems as shown in FIG.10

FIG.10 shows a destructive swap test circuit. For large quantum systems, the destructive swap test is more practical than the conventional swap test. In the conventional scheme, the control qubit must maintain a strong quantum correlation with all qubits that are been swapped. However, in the destructive swap test, each pair of qubits is measured separately, and the computation of purity only requires classical correlations between the measurement outcomes. In summary,

Theorem 3. The purity of a state p can be measured as an observable S, where Tr[p²] = Tr[p ® pSJ , such that the evaluation can be measured with a constant depth quantum circuit and a linear time classical post-processing.

Proof. The proof starts from the single-qubit case, where p acts on C².Let |'P⁺) == C|00> + |11) ), the projective measurement in the circuit shown in FIG.10 is given by the projectors that project to

where a and b take values from {0,1}, which are the result of the measurement on each subsystem. This is exactly the Bell basis, and hence the measurement is commonly known as the Bell measurement. The Bell basis can be rewritten as

where

are the Pauli operators:

I ⁼ °0,0’ ^ax ⁼ °l,0> °Z ⁼ °0,l’ °Y ⁼ °1,1 ■

This alternative form is derived from the identity, where for all operators A, B acting on C², A ® B |T⁺) - (AB^T>) ® |T⁺). (6)

As the result, the probability of obtaining measurement outcome (z_1;z₂) in the destructive swap test is given by

which uses the fact that Pauli operators are Hermitian. By applying equation (6) to the second copy of p, together with the definition of |'P⁺),

As the transpose is a Hermitian-preserving trace-preserving linear map on operators, it has a unique affine decomposition in terms of the Pauli operators: p^T Xz₁,z₂(-l)^Z1'^Z2 CTz₁,z₂ pCTz₁,z₂ (9)

Also, since transpose is an involution, we may interchange p with p^T in equation (9). As a result,

This indicates that the value of Tr[p²] can be obtained with a simple classical postprocessing, where after each measurement, (-1)^Z1'^Z2 is computed, whose mean is the value of [p²] . This procedure is equivalent to the evaluation of the expected value of the observable

This can be easily extended to multi-qubit systems, where the Bell measurements are performed pairwisely between the two copies of the same state. In this scenario, the measurement results are given by the Boolean vectorsz! and z₂ e B^ra, where the ith element of each vector corresponds to the measurement outcome of the ith pair of qubits. The multi-qubit Bell states are given by

^are the Bell states on ith qubit. The probability of each measurement outcome is given by

where a_Z1/Z2 == ®_;(j_{z. i/Z 2}.By applying identity (6) to all qubits simultaneously,

Since the transpose of a n-qubit state is equivalent to partial transpose each qubit,

05) where z_± ■ z₂ ■■= ^z^ ■ z_i2. The purity can be expressed as

That is, to measure the purity, one Bell measurement is applied to each pair of qubits. After the outcomes are obtained, the value (-1)^{Z1 Z2} is computed, whose mean is the value of Tr[p²]. As the Bell measurements are independent, all of them can be done simultaneously within two layers of quantum gates. The post-processing is a simple parity checking, whose complexity grows linearly with the number of qubits. The corresponding observables are

Theorem 4. The observable 5 in Theorem 3 is the swap operator between the two copies of p.

Proof. To check that S_n is indeed the swap operator between the two copies of p, we first notice that S_n can be expressed as two terms:

As Bell states

are orthogonal to each other, the two terms are (unnormalized) orthogonal projectors. As discussed in the proof of the Theorem 3, the Bell basis can be expressed as

By swapping the two subspaces

That is, is symmetric if

• z₂ - 0, and is anti-symmetric if it is 1 . By counting the dimension of the symmetric subspace, it is easy to show that those

with 0 form a complete basis of the symmetric subspace, and n_s := s the projector to the symmetric subspace. Similarly, those with

z₁ • z₂ = 1 form a complete basis of the anti-symmetric subspace, and the second term, denoted 11,₄ , is the projector to the anti-symmetric subspace. As a result, S_n - n_s - n_zl is the swap operator, which preserves the symmetric subspace and provides a -1 phase to the anti-symmetric subspace.

Theorem 5. Given an observable 0 defined for 2-copies o

f if(p) e M, and consider the expectation value £(p) with respect to a channel Tl that

p p The Haar average of E can be evaluated by measuring 0 on a single

initial state T e Jf®² , such that T can be prepared with a constant depth quantum circuit and a linear time classical pre-processing.

Proof. The average

where p^ = |ip) <i//| is distributed according to the Haar measure. As the input state

IVOIVO is symmetric at all instances, the Haar average of the input state is a projector state that

where n_s is invariant under swap. To prepare the projector state T, it is sufficient to find a basis of the projected space and then uniformly random sample the basis states Thus the preparation of state n_s/tr[n_s] is reduced to uniformly random sample and z₂ such that z_r • z₂ is even, and according to each pair of z_vz₂, prepare state

And the states I^^Jcan be prepared by applying the circuit shown in FIG.11 to classical states |z₁)|z₂) pairwisely.

FIG.11 shows a circuit that prepares Bell states. Similar to the Bell measurement, the preparation of Bell states only requires two layers of quantum gates. Although some subtle optimization may be used to sample the desired classical bits

and z₂, a naive rejective sampling can already uniformly sample the even bits in time linear to the number of qubits. This can be achieved by uniformly random sampling all bits in z_r and z₂ independently. Then resample the bits until z₁ ■ z₂ is even. The reject rate is less than 1/2 for all sizes of quantum circuits, as there 2^2n-1 + 2^n-1 pairs of z_r and z₂ that z_x ■ z₂ is even, and the total number of the configurations is 2²ⁿ.

Combining Theorem 3 and Theorem 5, we have the average linear entropy L of the output of a channel If:

This argument can be applied to the case where L is defined for a subsystem. Given the total space J-C is decomposed into

, the average linear entropy of the output of a channel U on subsystem A is given by

As the purity Tr[p²] is obtained from classical post-processing, the purity of each subsystem can be obtained simultaneously, with the same set of measured data. That is, for any partition of the qubits encoding p , all of the values of 7'rlYp'⁴)²] and Tr[(p^B)²] together with the Tr[p²] can be obtained with one shot of measurements on the quantum circuit. This property means that our cost function

can be evaluated with a single shot of measurements on the quantum circuit, and a linear time classical post-processing.

Parameter Shift Rule

Preliminary

The parameter shift rule is widely used in variational quantum circuits for gradientbased optimization. It outperforms the finite difference method, which estimate f(0) ~ A6+A9p-r(e) _w.th _a fj_{njte sma}|| nun-jber A0_;. In the finite difference method, 39/ A9; ¹ as the denominator A0_; is small, the difference of f is a small number, which is relatively sensitive to the noises. But, with the parameter shift rule, the gradient can be estimated directly, which is less sensitive to perturbations as long as the gradient is non-varnishing.

The usual parameter shift rule is based on the idempotent properties of Hermitian ■⁹i generators. Let us consider a single-parameterized unitary t/(0j) = e“^!T^H that is generated by an idempotent Hermitian operator H that H² = H. The parameter shift rule says Proposition 6. For all observables A and all initial states p, the derivative of the expectation value

Proof. The derivatives of the unitary operator and its conjugate are given by or p,

By matching the coefficients of the Taylor series, it can be shown that U(B_i~) - cos(0j/2) I — i sin(0j/2) H, and

— i[H, p] = — iHp + ipH

In summary, the derivative of the operator t t

The proposition is the direct result of equation (29).

Parameter Shift Rule for C_D Here we show that a similar parameter shift rule can be derived for our cost function C_D. Specifically, the derivative -^- C_D can be evaluated by four terms, each of which can be evaluated by the same quantum circuit for evaluating C_D . To simplify the notation in the following discussion, let U_e. denote the family of unitary channels that

The parameter shift rule of equation (29) can be rewritten as

Our cost function is defined by the expectation value of an observable 0

where the initial state p = T_A ® T_B, and W is a quantum circuit parameterized by 0.

In the design of the ansatz, for each parameter 0; , there is only one Pauli rotation gate that depends on 0_; in 1¥(0). We may split the circuit W into three parts as in FIG.12.

FIG.12 is a quantum circuit diagram showing the splitting of a parameterized quantum circuit into three parts. For each parameter 0_;, the parameterized circuit W can be split into three parts, where W = PU(Qi)Q. As there is only one Pauli rotation gate £7(0j) that depends on 0_£, all gates before t/(0_;) are collected in Q, and all gates after t/(0_;) are collected in P. For convenience, we put all gates parallel to t/(0_;) into Q.

Then

Let W_Q and ll"_e. denote the unitary channels corresponding to t/(0j) ® H and H ® t/(0i) respectively. And let P,Q denote the unitary channels corresponding to P®²andQ®². Then the unitary evolution can be written as the composition of 4 unitary channels:

According to the product rule of derivatives,

As the Pauli rotations 67(0j) are generated by idempotent operators and satisfies the parameter shift rule of equation 29, the same property holds for lf'_e. and U"₉.. In conclusion, the derivative of our cost function is

This is the sum of 4 terms, each of which can be evaluated by the same method of evaluating C_D(W) with a "parameter shift".

Relating decoupling cost to the gate fidelity

Characteristics of the gate fidelity The closeness of approximating U with U_A ® U_B is measured by the gate fidelity between U and U_A ® U_B, which is defined by

This Haar average can be contracted to a single quantity that

where d is the dimension of the global Hilbert space. Here {U_A 0 U_B, U) := ® U_B)^fU] denotes the Hilbert-Schmidt inner product between U_A 0 U_B and U.

Note that, the average fidelity is maximized when (U_A 0 U_B, U) is maximized. Let 'll be the unitary channel that 1/(-) =

and let U_A and U_B be the corresponding channels for U_A and U_B. Denote the Choi-Jamiolkowski matrix of a channel 'll by J^u. Then the inner product can be expressed as

As the Choi rank of unitary channels are all 1, we may rewrite the Choi operators as the outer products

Then

As \ip) is a vector in the Hilbert space, there exists a Schmidt decomposition such that

where Z_k p_k = (v|r|^) =

according to the normalization condition for quantum channels. As the result, the maximum value of \{U_A ® U_B, U)\² is given by

Let q_{A k} ■= (ijj/i l^/<) and q_{B k} ■= (IJJB I't’/cX ^we have

According to the Cauchy-Schwartz inequality,

KIP/,III^I)|² < llqJli ll^lli (44)

Given |^_fc)and I4>_fc) are orthonormal bases, llq^Hi =

= d_B

As above inequalities hold for all 14^), |I|J_B),

where the infinity norm HpH^ equals to the maximum Schmidt coefficient of \ip). The equality holds when \ip_A} and \ip_B} are proportional to the |^_fe) and I4>_fc) that corresponding to the maximum p_k. However, this might not be achievable as |^_fc) and I4>_k) do not necessarily correspond to unitary operators.

In summary, for all U_A ® U_B, the gate fidelity is bounded by

1 1

— - < F(U_A ® U_B,U) < — — (llpIL + 1) < 1 d + 1 a + 1

(46) In other words

To fully understand the property of separability, we would like to explore the gate fidelity with respect to an alternative set. Without loss of generality, assuming the dimension of the system A, d_A, is not larger than d_B. Then we may extend the Hilbert space of A to A , whose dimension is the same as B, such that A = A ® C, where C is a d_B - d_A dimensional Hilbert space. We may extend U to construct a unitary operator in A⁺ , such that U = U ® /_B0C. As the dimensions of A⁺ and B are the same, we may define the swap operator S between A⁺ and B. Now, we will focus on the gate fidelity

Let {q_k} be the Schmidt coefficients of the Choi operator with respect to SU .

Then, with similar arguments as above, we have

Since dj > d_Ad_B = d by assumption,

This quantity would be useful in the analysis of the decoupling cost.

Characteristics of the decoupling cost In this section, we show that the maximum gate fidelity F_max related to the decoupling cost function C_D, such that

The evaluation of C_D requires two copies of a target unitary U, whose acts on the system A 0 B. We may label the systems as 4_1; A₂,B_lt B₂, where one unitary acts on A_r 0 B_lt and the other acts on A₂ ® B₂. In terms of the Choi-Jamiolkowski operator, the average linear entropy, denoted L, is given by

where S_A (S_B) is the swap operator between the output systems A_t and A₂ , (or B systems respectively), and the identity operators ll_z1, ll_B are also defined on the system A_r 0 A₂ or B_r 0 B₂ respectively.

This equality comes from Theorem 2, where the partial transposes from the formula are omitted, as T_k,ll_fe and 5_fc are all invariant under the transpose. In this Choi representation, we need to distinguish the output systems and input systems, and thus there are 8 subsystems in the formula in total. We may label the subsystems as A^A^B^Bz for the input systems, and A_i ,A₂ ,B_r ,B₂ for the output systems. Note that, the initial state r_k - + 1), and the trace of the projector tr[P₅] is

for the pair of systems (B_1ZB₂), where d_A, d_B are the dimension of system A^Bt respectively. Then each of the two terms in 2(1 - L) is divided into 4 terms:

For convenience, we label each term in the parenthesis as P_k in the order presented in equation (53), where k ranges from 1 to 8. That is

By changing the order of the tensor products and grouping the swap operators, we may rewrite each P_k in the form of tr[S_a 0 /p(/a₁,p₁ 0 /«₂,p₂)] for some partitioning (a, p) of the four pairs the systems (Ap^) , (BI, B₂) , This formula is equivalent to tr [s_atr_p [/_{ai P1}] 0 trp [/ct₂,p₂] ]

Given j^u is a rank-1 operator, we can write it in its Schmidt decomposition \I/J) = l<P/c)_a ® I Efc )p- After the partial trace of the p systems,

Then, by applying the swap and the trace

This result indicates that each term of the 8 terms is the 2-norm of the Schmidt coefficients of \tp} for some partitioning (a, p) of the system J¹¹ acting on. That is saying, every term P_k is positive and upper bounded by the inner product of \t[P), which is d.

Now, we need to analyze each of the terms individually. First notice that each of the terms is the 2-norm of the Schmidt coefficients of \ip), which only depends on the partitioning of the system. Specifically,

As the result, by interchanging the subsystems a and p, the 8 terms can be grouped into 4 terms:

P₂ is strongly related to the gate fidelity between U and U_A ® U_B. For P₂, the partition (a, p) matches the partition (A B) in the analysis of the fidelity as in equation (41 ), which means P₂ = p₂.

P_± and P₅ are equivalent to the expectation value of S_A 01_B and I_A 0 S_B respectively with completely mixed state as input. That is

Pi = tr[(^ ® l_B)W) ® W)]

where d_A and d_B are the dimensions of the subsystems A_t and B; respectively. Similarly, P₅ = d_Ad_B

In summary,

As the dimension of the whole system A 0 B is d = d_Ad_B, we have

Which means

Now, we will substitute the definition C_D = -^r^L to equation (63), where we assume dA ¹ d_A

^A < d_B u. Let then

d² + d - P₃-KC_Dd(d + 1 + d_A + d_B) (64)

Compared to the gate fidelity that

We may organize the terms to get

and

As d = d_Ad_B > d_A by assumption, we have,

and the above equation can be further simplified to

The quantity of P₃ here is related to the extended unitary u' as defined in equation (48).

where the first equality comes from the fact that both S_z1- and V are only supported on A ® A and C ® C, but not A ® C and C ® A, and hence the J¹ ® /^Ib®^c and J^1B®^C ® /'^u terms are zero. Observe that, similar to the analysis of P₂, the P₃ is the 2-norm of the Schmidt coefficients

where the output systems A⁺ and B are swapped in the first equality. In summary, we have (dj - d) d) (71 )

Where F£_ax is the maximum gate fidelity of the extended unitary SU' as defined in equation (48). Substituting this to equation (68), we have

In the case d_A = d_B, we have d = d_B, and

and the extended unitary U is the same as U. It shows that the maximum gate fidelities F_max and F£_ax are jointly bounded by 1 - C_D.

To obtain a neater but looser bound, we start from equation (68), then observe that P₃ is positive, and F_max < 1 such that

(74)

By combining the factors, we have

In the present disclosure, we have outlined a variational means to decouple complex quantum circuits. This is enabled by the proposal of an efficiently measurable cost function CD, that allows us to identify suitable pre- and post-processing circuits to decouple a 2n-qubit unitary to a tensor product of two n-qubit unitary operators. Through the recursive application, we illustrated how variational decoupling enhances existing approaches to circuit compilation - a key task in the design of quantum circuits in both near-term of future universal quantum computers. In benchmarks, variational decoupling enabled us to compile circuits realization of arbitrary two and four-qubit quantum gates to significantly higher fidelity. By setting the allowed depths of preprocessing and post-processing circuits, our decoupling method can be adapted to discovering circuits of varying expressivity, ranging from very shallow circuits for NISQ devices to deep circuits that represent general unitary operations.

Whilst the foregoing description has described exemplary embodiments, it will be understood by those skilled in the art that many variations of the embodiments can be made within the scope and spirit of the present invention.

Claims

1 . A method of decomposing a quantum circuit, the method comprising: dividing the quantum circuit into a first sub-circuit and second sub-circuit; determining a pre-processing unitary operator and a post-processing unitary operator which approximate local evolutions on the first sub-circuit and second subcircuit; determining gate decompositions of the pre-processing unitary operator and the post-processing unitary operator: and using the gate decompositions of the pre-processing unitary operator and the post-processing unitary operator to determine a gate decomposition of the quantum circuit.

2. The method according to claim 1 , further comprising decomposing the first subcircuit and the second sub-circuit into smaller sub-circuits.

3. The method according to claim 2, comprising limiting a gate depth of the quantum circuit.

4. The method according to any one of the preceding claims, wherein determining gate decompositions of the pre-processing unitary operator and the post-processing unitary operator comprises parametrizing the pre-processing unitary operator and the post-processing unitary operator and minimizing a cost function.

5. The method according to claim 4, wherein the cost function is a global cost function.

6. The method according to claim 4, wherein the cost function is a sum of local cost functions each restricted to a respective qubit.

7. The method according to any one of claims 4 to 6, wherein determining gate decompositions of the pre-processing unitary operator and the post-processing unitary operator comprises a gradient descent method.

8. The method according to any one of claims 4 to 7, wherein determining decompositions of the pre-processing unitary operator and the post-processing unitary operator comprises controlling a quantum processing unit to input entangled qubit pairs into the quantum circuit and measuring the cost function and its gradient using a destructive swap test.

9. The method according to claim 8, wherein the entangled qubit pairs are entangled via a CNOT gate.

10. The method according to claim 8 or claim 9, wherein the destructive swap test comprises applying CNOT and Hadamard gates.

11. A computer readable medium storing processor executable instructions which when executed on a processor cause the processor to carry out a method according to any one of claims 1 to 10.

12. A system for decomposing a quantum circuit, the system being configured to: divide the quantum circuit into a first sub-circuit and second sub-circuit; determine a pre-processing unitary operator and a post-processing unitary- operator which approximate local evolutions on the first sub-circuit and second subcircuit; determine gate decompositions of the pre-processing unitary operator and the post-processing unitary operator; and use the gate decompositions of the pre-processing unitary operator and the post-processing unitary operator to determine a gate decomposition of the quantum circuit.

13. The system according to claim 12, being further configured to decompose the first sub-circuit and the second sub-circuit into smaller sub-circuits.

14. The system according to claim 13, being further configured to limit a gate depth of the quantum circuit.

15. The system according to any one of claims 12 to 14, being further configured to determine gate decompositions of the pre-processing unitary operator and the postprocessing unitary operator by parametrizing the pre-processing unitary operator and the post-processing unitary operator and minimizing a cost function.

16. The system according to claim 15, wherein the cost function is a global cost function.

17. The system according to claim 15, wherein the cost function is a sum of local cost functions each restricted to a respective qubit.

18. The system according to any one of claims 15 to 17, being further configured to determine gate decompositions of the pre-processing unitary operator and the postprocessing unitary operator by a gradient descent method.

19. The system according to any one of claims 15 to 18, further comprising a quantum processing unit and being further configured to determine decompositions of the pre-processing unitary operator and the post-processing unitary operator by controlling the quantum processing unit to input entangled qubit pairs into the quantum circuit and measuring the cost function and its gradient using a destructive swap test.

20. The system according to claim 19, wherein the entangled qubit pairs are entangled via a CNOT gate.

21. The system according to claim 19 or claim 20, wherein the destructive swap test comprises applying CNOT and Hadamard gates.